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OVERVIEW
In today’s lecture we will study fixed income
securities. We will look at:
Bond characteristics;
Bond pricing;
Yields; and,
The term structure of interest rates.
2BONDCHARACTERISTICS
Recall that bonds are debt instruments with
contractually agreed upon terms for:
Interest payments; and,
The repayment of the debt.
3BONDCHARACTERISTICS
Recall that bonds are debt instruments with
contractually agreed upon terms for:
Interest payments; and,
The repayment of the debt.
Remember also that, while shares can only be
issued by corporations, debt instruments can be
issued by many different types of entities including
governments (treasury bonds and notes) and
corporations (corporate bonds)
3BONDCHARACTERISTICS
categorised on the basis of whether or not they
make separate interest payments.
Zero-coupon bonds (or discount securities):
Do not pay coupons to the lender;
Involve the borrower paying the lender the
face (par) value at a pre-defined maturity date;
Trade at a discount, or a price less than their
face value. The difference between the issue
price and the face value represents the
interest accruing to the holder over the
instrument over its life. 4BONDCHARACTERISTICS
coupon-paying bonds
borrower agreeing to pay the lender
periodic payments known as coupon
payments, usually made on a semi-annual
basis over the life of the instrument; and,
The face value (or par value) of the
instrument at a pre-defined maturity date.
5CORPORATEBONDS
Callable bonds allow the issuer to buy them
back from the holder at a pre-determined call
price prior to maturity
Convertible bonds give holders option to
exchange each bond for a specified number of
shares of the firm’s stock
Puttable bond gives holder option to exchange
for par value at some date or to extend for a
given number of years
Floating-rate bond has interest rate that is reset
periodically according to a specified market rate 6BONDPRICINGAND
DEFAULTRISK
While bonds promise a level of income, it is not
riskless. The likelihood of issuers not making
promised payments is:
Known as default or credit risk;
Monitored by rating agencies such as Standard &
Poor’s Corporation ("S&P") and Moody’s Investor
Services ("Moody’s")
7BONDRATINGS
Ratings are made based on the level of, and
trends in, issuers’ financial ratios such as
profitability, leverage and liquidity.
The maximum rating issued by S&P and
Moody’s is AAA and Aaa respectively;
Bonds receiving relatively high ratings (S&P
rating at least BBB, Moody’s at least Baa) are
classed as investment-grade bonds;
Bonds with ratings below the thresholds are
classed as speculative-grade or junk bonds
As credit ratings decrease, both the likelihood
of default and the bond yield increases. 8BONDRATINGS
9BONDRATINGS
10BONDINDENTURES
Bonds are issued with indentures, contract to
protect the purchaser against the risk of default
Sinking fund calls for the issuer to periodically
repurchase some proportion of the outstanding
bonds prior to maturity
Subordination clauses restrict the amount of
additional borrowing by the firm
Dividend restrictions limit the payment of
dividends by firms
Collateral is a particular asset that the
bondholders receive if the firm defaults 11BONDPRICING
The price of a zero coupon bond maturity in T
periods given an interest (discount) rate of r is
simply calculated as the present value of its par
value, or:
par value
𝑝𝑟𝑖𝑐𝑒 =
(1 + 𝑟)
𝑇
12BONDPRICING
The price of a zero coupon bond with exactly 30
years to maturity given it has a face value of $1,000
and assuming a market interest rate is 8% is simply
𝑝𝑟𝑖𝑐𝑒 = = 99.38
1000
(1 + 0.08)
30
13BONDPRICING
Assuming a constant interest (discount) rate, r, the
price of a bond maturing in exactly T coupon
periods can be calculated as follows:
Where coupons are generally paid semi-annually as a percentage of
the par value and both coupon and interest rates are quoted as per
annum rates compounded semi-annually.
𝑝𝑟𝑖𝑐𝑒 = (1
−
)coupon + par value
1
𝑟
1
(1 + 𝑟)
𝑇
1
(1 + 𝑟)
𝑇
14BONDPRICING
On Jan 2, 2018 and you purchase a bond that has a
face value of $1,000, pays a six-monthly coupon at
a rate of 6% pa and matures on Jan 1, 2021.
Assuming a market interest rate of 4% p.a.:
There will be 6 coupon payments of $30 each in
July 2018, Jan 2019, July 2019, Jan 2020, July 2020,
Jan 2021;
The face value of $1000 is payable on Jan 1, 2021;
The market interest rate is r=4% p.a., or 2% per
half-year
15BONDPRICING
Based on this information, the price of the bond is:
the bond is selling at a premium, or for an amount
that is greater than its face value.
𝑝𝑟𝑖𝑐𝑒 = (1 − )30 + 1000
1
0.02
1
(1.02)
6
1
(1.02)
6
𝑝𝑟𝑖𝑐𝑒 = 1, 056.01
16BONDPRICING
Notice:
This approach can only be used to calculate the
price immediately after a coupon has been paid;
If, instead, a bond is purchased between coupon
dates, then the price should reflect the time that
has passed since the last coupon payment.
17BONDPRICING
What is the price of this bond on Feb 3, 2018,
assuming 182 days between coupon payments?
As there have been 33 days since the last coupon
payment, the bond price is simply
1056.01 ⋅ 1.02 = 1059.81
33
182
18BONDPRICESOVERTIME
Extending the example on the preceding slide even
further:
On Jan 2, 𝑃1 = 1, 056.01;
On Feb 3, 𝑃2 = 1, 059.81;
On July 1, 𝑃3 = 1, 077.14;
On July 2, 𝑃4 = 1, 047.14 That is, as interest is
accrued every day, the bond price increases
within each coupon period. However, after the
coupon payment, the bond price will fall by the
amount of the coupon.
19BONDPRICESOVERTIME
More generally, bond prices will move towards par
value over time.
20BONDPRICESANDINTERESTRATES
bond prices at different interest rates (8% coupon, paid
semiannually):
21BONDPRICESANDINTERESTRATES
22BONDPRICESANDINTERESTRATES
Inverse relationship between price and interest
rate is a central feature of fixed-income
securities
Interest rate fluctuations represent the main
source of risk in the fixed-income market
The price curve is convex and becomes flatter at
higher interest rates
The longer the maturity of the bond, the more
sensitive the bond’s price to changes in market
interest rates
23BONDYIELDS:YIELDTOMATURITY
Yield to maturity (YTM) is the interest rate that
makes the present value of a bond’s payments
equal to its price
Interpreted as a measure of the average rate
of return that will be earned on a bond if it is
bought now and held until maturity
To calculate YTM, solve the bond price equation
for the interest rate given the bond’s price
implicitly assumes all coupon payments are
reinvested at the YTM.
24YIELDTOMATURITY
YTM = 2%
1, 056.01 = (1
−
)30 + 1000
1
𝑌 𝑇𝑀
1
(1 + 𝑌 𝑇𝑀)
6
1
(1 + 𝑌 𝑇𝑀)
6
25YIELDTOMATURITY
Given the coupon rate and maturity, there is an
inverse relationship between bond price and
YTM;
When interest rate is constant over the life of the
bond, YTM = interest rate; and,
Bonds are often quoted in terms of YTM.
26REALIZEDYIELD
As a bond’s YTM is based on its current price, it
changes over time.
A bond’s YTM is not the actual realized yield
because in most cases, the reinvestment rate ≠
YTM;
The realized yield is based on the actual
reinvestment rate in each period and the total
cash received at maturity.
27CURRENTYIELD
Current yield is the bond’s annual coupon
payment divided by its price.
In the example provided earlier, the bond’s
current yield is 60/1056.01 or 5.68%.
28YTMVSCURRENTYIELD
Yield to maturity
Bond’s internal rate of return
Interpreted as compound rate of return over
life of the bond assuming all coupons can be
reinvested at that yield
Proxy for average return
Premium bonds: Coupon rate > Current yield >
YTM
Discount bonds: Coupon rate < Current yield <
YTM
29YTMANDDEFAULTRISK
Default premium is a differential in promised
yield that compensates the investor for the risk
inherent in purchasing a corporate bond that
entails some risk of default
30YTMANDDEFAULTRISK
31CALLABLEBONDS
The YTM assumes that a bond will be held until maturity.
This is not a realistic assumption for callable bonds, which
might be retired before their maturity date.
If interest rates fall, PV of bond's scheduled
payments, and the price of a straight bond can
rise considerably.
If the bond is called, the bondholder will only
receive the call price (e.g., 100% of par), not the
market price.
If call price is less then PV of scheduled
payments, issuer may call the bond back
32YIELD-TO-CALL
We often calculate a yield to call rather than YTM
for callable bonds
The yield to call replaces time until maturity with
time until call and par value with call price in the
YTM calculation.
Most bonds are issued with initial call protection
periods.
Deeply discounted bonds (relative to call price)
offers implicit call protection
33YIELD-TO-CALL
Low interest rates
The price of the callable bond is flat since the
risk of repurchase or call is high
High interest rates
The price of the callable bond converges to
that of a normal bond since the risk of call is
negligible
34BONDPRICES:CALLABLEANDSTRAIGHTDEBT
35TERMSTRUCTUREOF
INTERESTRATES
We have assumed that the same interest rate would be
used to discount cash flows, regardless of when they are
expected to be received.
This is unrealistic given shorter-term securities
commonly have lower yields than longer-term
securities;
The structure of interest rates used when
discounting cash flows of varying maturities is
called the term structure of interest rates.
36YIELDCURVE
Plotting the YTM as a function of time to maturity
yields a graphical depiction of the relationship
known as the yield curve:
The pure yield curve describes the relationship
between YTM and time to maturity for stripped
or zero-coupon treasuries;
The on-the-run yield curve describes the
relationship between YTM and time to maturity
for newly issued coupon-paying bonds selling at /
close to par.
37YIELDCURVE
38VALUINGCOUPONPAYINGBOND
If bonds with different maturities have different yields,
how do we value coupon-paying bonds, which make
payments at numerous times throughout their lives?
View each of the bond’s cash flows as a zero-
coupon bond; and,
Discount it at the yield appropriate to its
maturity.
39VALUINGCOUPONPAYINGBOND
Imagine the YTMs on zero-coupon bonds with 1, 2,
3 and 4 years to maturity are 4%, 5%, 6% and 7%
respectively. A bond with a 1,000 dollar face value
and 4 years to maturity paying annual coupons at a
rate of 8% p.a. would be valued at 1,040.58,
calculated as follows:
+ + + = 1, 040.58
80
1.04
80
1.05
2
80
1.06
3
80
1.07
4
40FUTUREINTERESTRATES
41FUTUREINTERESTRATES
In equilibrium, both strategies must provide the
same return, that is the short rate in year 2 (r2)
must be:
(1 + 𝑦2)
2
= (1 + 𝑦1)(1 + 𝑟2)
(1 + 𝑦2)
2
1 + 𝑟2 =
(1 + 𝑦1)
42FUTUREINTERESTRATES
43FORWARDRATES
We cannot determine the short rate that will
prevail in future periods. Instead, can only
forecast it;
Hence we replace the short rate in the last
equation with the forward rate, 𝑓𝑡, as an
"estimate" of the future short rate in year t
(1 + 𝑦𝑛)
𝑛
1 + 𝑓𝑛 =
(1 + 𝑦𝑛
−
1)
𝑛
−
1
𝑓𝑛 is the 1-year rate after (n-1) years.
44INTERESTRATEUNCERTAINTYANDFORWARDRATES
When investors are risk aversed and the short rate for
period two is not known today, each investor wants to be
compensated for taking on a security which does not
match investment horizon.
a 1-year investor demands a premium to invest
in 2-year bonds
(1 + 𝑦2)
2
> (1 + 𝑦1)(1 + 𝔼(𝑟2))
𝔼(𝑟2) < 𝑓2
a 2-year investor demands a premium to invest
in 1-year bonds
45THEORIESOFTERMSTRUCTURE
Theories of term structure theories examine
determinants of yields at different maturities and
the implications for future short rates.
The Expectations Hypothesis;
The Liquidity Preference Hypothesis;
The Market Segmentation Hypothesis
46EXPECTATIONSHYPOTHESISTHEORY
Yield curve reflects the market’s expectations of
future interest rates
The forward rate is an unbiased estimate of the
future short rate, that is, 𝔼(𝑟𝑛) = 𝑓𝑛 ;
Implicitly, investors are risk neutral
47LIQUIDITYPREFERENCETHEORY
upward bias built into the long-term rates because
of the liquidity premium
more short-term investors than long-term
investors, meaning long term bonds have lower
liquidity than short-term bonds;
To compensate buyers of long-term bonds for
the lower liquidity, long-term bonds must offer a
higher return, 𝑓2 > 𝔼(𝑟2)
48MARKETSEGMENTATIONHYPOTHESISTHEORY
Finally, the Market Segmentation Hypothesis
argues that:
Borrowers and lenders have strong preferences
for particular maturities. Consequently, they do
not hold or issue bonds at other maturities;
Debt markets at different maturities are not
linked, that is, they are segmented; and,
The yield at a particular maturity is determined
purely by supply and demand for bonds at that
maturity.
49INTERPRETINGTHETERM
STRUCTURE
Yield curve reflects expectations of future short
rates, but also reflects other factors such as
liquidity premiums
An upward sloping curve could indicate:
Rates are expected to rise; and/or
Investors require large liquidity premiums to
hold long term bonds
50INTERPRETINGTHETERM
STRUCTURE
The yield curve is a good predictor of the
business cycle
Long-term rates tend to rise in anticipation of
economic expansion
Inverted yield curve may indicate that interest
rates are expected to fall and signal a
recession
