Bond Valuation, Duration and Convexity Flashcards
Why bonds are important
Bonds allow long-term borrowing from financial
markets
* Bank loans alternative to bond markets
- Bank credit by far is largest source of finance for the US
& UK firms
Bond
Security that obligates the issuer to make specified payments to
the bondholder.
maturity date
Final repayment (of principal) date
term
The time remaining until the repayment date
face value
Payment at the maturity of the bond.
coupon
The interest payments made to the bondholder.
coupon rate
Annual interest payment, as a percentage of face value.
coupon payment equation
(coupon rate * face value)/ number of coupon payments per year
zero-coupon bond
– Does not make coupon payments
– Always sells at a discount (a price lower than face value),
so they are also called pure discount bonds
– Treasury Bills are U.S. government zero-coupon bonds
with a maturity of up to one year.
Yield to Maturity
The discount rate that sets the present value of the
promised bond payments equal to the current market
price of the bond
Risk-Free Interest Rates
– A default-free zero-coupon bond that matures on date n
provides a risk-free return over the same period.
– Thus, the Law of One Price guarantees that the risk-free
interest rate equals the yield to maturity on such a bond.
– Risk-Free Interest Rate with Maturity n.
Premium
– Bond price is greater than the face value
– YTM < coupon rate
– An investor will earn a return from receiving the coupons, but this return
will be diminished by receiving a face value less than the price paid for the
bond.
time and bond prices
Holding all other things constant, a bond’s yield to
maturity will not change over time.
* Holding all other things constant, the price of
discount or premium bond will move toward par
value over time.
* If a bond’s yield to maturity has not changed, then
the I R R of an investment in the bond equals its yield
to maturity even if you sell the bond early.
interest rate sensitivity 1-5
- Bond prices and yields are inversely related
- An increase in a bond’s yield to maturity results
in a smaller price change than a decrease in
yield of equal magnitude (convex) - The long-term bond price is more sensitive to
interest rate changes than short-term bond
price - The lower yield bond price is more sensitive to
interest rate changes than the higher yield bond
price - The lower coupon bond price is more sensitive
to interest rate changes than the higher coupon
bond price
Inverse relationship between price and yield
➢ As interest rates and bond yields rise, bond prices fall
➢ As interest rates and bond yields fall, bond prices rise
Macaulay’s duration
equals the weighted average of
the times to each coupon or principal payment
Duration = Maturity for zero-coupon bonds
* Duration < Maturity for coupon bonds
Duration
The sensitivity of a bond’s price to changes in
interest rates is measured by the bond’s
duration.
– Bonds with high durations are highly sensitive to interest
rate changes.
– Bonds with low durations are less sensitive to interest rate
changes.
Modified duration.
Tell us how much a bond’s price changes (in percent) for a
given change in yield.
Money (Dollar) duration.
Tell us how much a bond’s price changes (in dollars) for a
given change in yield.
modified duration equation
Modified Duration (D∗) =
D / (1 + yield)
duration rule 1
The (Macaulay) duration of a zero-coupon bond equals its
time to maturity
duration rule 2
The (Macaulay) duration of a perpetuity is equal to:
(1+y)/y
𝑁𝑜𝑡𝑒: 𝑦 = 𝑌𝑇𝑀
duration rule 3
Holding other factors constant, a bond’s (modified/money)
duration generally increases with its time to maturity
durations rule 4
Holding other factors constant, a bond’s (modified/money)
duration is higher when the bond’s yield to maturity is lower
duration rule 5
Holding other factors constant, a bond’s (modified/money)
duration is higher when the coupon rate is lower
corporate bonds
- issued by corporations
- risk of default
certain default of corporate bonds
When computing the yield to maturity for a bond with
certain default, the promised rather than the actual cash
flows are used.
determinants of bond safety
– Coverage ratios
– Leverage (e.g., debt-to-equity) ratios
– Liquidity ratios
– Profitability ratios
– Cash flow-to-debt ratio
default spread
– Also known as Credit Spread
– The difference between the yield on corporate bonds and
Treasury yields
Credit default swap (CDS)
– Acts like an insurance policy on the default risk of a
bond or loan
– Allows lenders to buy protection against default risk
– Risk structure of interest rates and CDS prices ought to
be tightly aligned
– CDS contracts trade on corporate as well as sovereign
debt
Collateralized Debt Obligations
(CDOs)
– Major mechanism to reallocate credit risk in the fixed-
income markets
– To create a CDO, a legally distinct entity to buy/resell a
portfolio of bonds must be established
* Structured Investment Vehicle (SIV)
– Loans are pooled together and split into tranches, where
each tranche is given a different level of seniority in terms
of its claims on the underlying loan pool
– Mortgage-backed CDOs were an investment disaster in
2007-2009
sovereign bonds
bonds issued by national governments
- not default-free
sovereign bonds and default risk– Foreign currency debt
- Default occurs when foreign government borrows
dollars - If crisis occurs, governments may run out of taxing
capacity and default - Affects bond prices, yield to maturity
Sovereign Bonds and Default Risk
– Own currency debt
- Less risky than foreign currency debt
- Governments can print money to repay bonds
Sovereign Bonds and Default Risk
– Eurozone debt
- Can’t print money to service domestic debts
- Money supply controlled by European Central Bank
duration rule is good approx only for…
small changes in bond yields
Why is duration more accurate for small changes in yield than
for large changes?
Because duration is a linear approximation of a curvilinear (or convex)
relation:
* Duration treats the price/yield
relationship as a linear.
* Error is small for small ∆𝑟
* Error is large for large ∆𝑟
* The error is larger for yield
decreases.
* The error occurs because of
convexity.
Convexity (of bond)
rate of change of
slope in Price-YTM curve as fraction of
bond price
Why Do Investors Like Convexity?
- Bonds with greater curvature gain more in price
when yields fall than they lose when yields rise
– The more volatile interest rates, the more attractive this
asymmetry - Investors must pay higher prices and accept lower
yields to maturity on bonds with greater convexity
Passive Bond Management
Passive managers take bond prices as fairly set and
seek to control only the risk of their fixed-income
portfolio
Two classes of passive management:
– Indexing strategy
– Immunization techniques
Bond-Index Funds
Similar to stock market indexing
– Idea is to create a portfolio that mirrors the
composition of an index that measures the broad
market
Challenges in construction of bond-index funds
- Very difficult to purchase each security in the index in
proportion to its market value - Many bonds are very thinly traded
- Difficult rebalancing problems
Cash flow matching
is a form of immunization that
requires matching cash flows from a bond portfolio
with those of an obligation
dedication strategy
– Manager selects either zero-coupon of coupon bonds with
total cash flows in each period that match a series of
obligations
– Once-and-for-all approach to eliminating interest rate risk
Active Bond Management:
Sources of Potential Profit (5)
- Substitution swap – exchange of one bond for another
more attractively priced bond with similar attributes - Intermarket spread swap – switching from one segment of
the bond market to another (e.g., from Treasuries to
corporates) - Rate anticipation swap – switch made between bonds of
different durations in response to forecasts of interest rates - Pure yield pickup swap – moving to higher-yield, longer-
term bonds to capture the liquidity premium - Tax swap – swapping two similar bonds to capture a tax
benefit
Horizon analysis
involves forecasting the realized
compound yield over various holding periods of
investment horizons
