Bond Valuation, Duration and Convexity

Question 1

Why bonds are important

Answer 1

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

Question 2

Bond

Answer 2

Security that obligates the issuer to make specified payments to
the bondholder.

Question 3

maturity date

Answer 3

Final repayment (of principal) date

Question 4

term

Answer 4

The time remaining until the repayment date

Question 5

face value

Answer 5

Payment at the maturity of the bond.

Question 6

coupon

Answer 6

The interest payments made to the bondholder.

Question 7

coupon rate

Answer 7

Annual interest payment, as a percentage of face value.

Question 8

coupon payment equation

Answer 8

(coupon rate * face value)/ number of coupon payments per year

Question 9

zero-coupon bond

Answer 9

– 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.

Question 10

Yield to Maturity

Answer 10

The discount rate that sets the present value of the
promised bond payments equal to the current market
price of the bond

Question 11

Risk-Free Interest Rates

Answer 11

– 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.

Question 12

Premium

Answer 12

– 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.

Question 13

time and bond prices

Answer 13

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.

Question 14

interest rate sensitivity 1-5

Answer 14

1. Bond prices and yields are inversely related
2. An increase in a bond’s yield to maturity results
   in a smaller price change than a decrease in
   yield of equal magnitude (convex)
3. The long-term bond price is more sensitive to
   interest rate changes than short-term bond
   price
4. The lower yield bond price is more sensitive to
   interest rate changes than the higher yield bond
   price
5. The lower coupon bond price is more sensitive
   to interest rate changes than the higher coupon
   bond price

Question 14

Inverse relationship between price and yield

Answer 14

➢ As interest rates and bond yields rise, bond prices fall
➢ As interest rates and bond yields fall, bond prices rise

Question 15

Macaulay’s duration

Answer 15

equals the weighted average of
the times to each coupon or principal payment
Duration = Maturity for zero-coupon bonds
* Duration < Maturity for coupon bonds

Question 16

Duration

Answer 16

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.

Question 17

Modified duration.

Answer 17

Tell us how much a bond’s price changes (in percent) for a
given change in yield.

Question 18

Money (Dollar) duration.

Answer 18

Tell us how much a bond’s price changes (in dollars) for a
given change in yield.

Question 19

modified duration equation

Answer 19

Modified Duration (D∗) =
D / (1 + yield)

Question 20

duration rule 1

Answer 20

The (Macaulay) duration of a zero-coupon bond equals its
time to maturity

Question 21

duration rule 2

Answer 21

The (Macaulay) duration of a perpetuity is equal to:
(1+y)/y
𝑁𝑜𝑡𝑒: 𝑦 = 𝑌𝑇𝑀

Question 22

duration rule 3

Answer 22

Holding other factors constant, a bond’s (modified/money)
duration generally increases with its time to maturity

Question 23

durations rule 4

Answer 23

Holding other factors constant, a bond’s (modified/money)
duration is higher when the bond’s yield to maturity is lower

Question 24

duration rule 5

Answer 24

Holding other factors constant, a bond’s (modified/money) duration is higher when the coupon rate is lower

Question 25

corporate bonds

Answer 25

- issued by corporations - risk of default

Question 26

certain default of corporate bonds

Answer 26

When computing the yield to maturity for a bond with certain default, the promised rather than the actual cash flows are used.

Question 27

determinants of bond safety

Answer 27

– Coverage ratios – Leverage (e.g., debt-to-equity) ratios – Liquidity ratios – Profitability ratios – Cash flow-to-debt ratio

Question 28

default spread

Answer 28

– Also known as Credit Spread – The difference between the yield on corporate bonds and Treasury yields

Question 29

Credit default swap (CDS)

Answer 29

– 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

Question 30

Collateralized Debt Obligations (CDOs)

Answer 30

– 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

Question 31

sovereign bonds

Answer 31

bonds issued by national governments - not default-free

Question 32

sovereign bonds and default risk– Foreign currency debt

Answer 32

* 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

Question 33

Sovereign Bonds and Default Risk – Own currency debt

Answer 33

* Less risky than foreign currency debt * Governments can print money to repay bonds

Question 34

Sovereign Bonds and Default Risk – Eurozone debt

Answer 34

* Can’t print money to service domestic debts * Money supply controlled by European Central Bank

Question 35

duration rule is good approx only for...

Answer 35

small changes in bond yields

Question 36

Why is duration more accurate for small changes in yield than for large changes?

Answer 36

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.

Question 37

Convexity (of bond)

Answer 37

rate of change of slope in Price-YTM curve as fraction of bond price

Question 38

Why Do Investors Like Convexity?

Answer 38

* 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

Question 39

Passive Bond Management

Answer 39

Passive managers take bond prices as fairly set and seek to control only the risk of their fixed-income portfolio

Question 40

Two classes of passive management:

Answer 40

– Indexing strategy – Immunization techniques

Question 41

Bond-Index Funds

Answer 41

Similar to stock market indexing – Idea is to create a portfolio that mirrors the composition of an index that measures the broad market

Question 42

Challenges in construction of bond-index funds

Answer 42

* Very difficult to purchase each security in the index in proportion to its market value * Many bonds are very thinly traded * Difficult rebalancing problems

Question 43

Cash flow matching

Answer 43

is a form of immunization that requires matching cash flows from a bond portfolio with those of an obligation

Question 44

dedication strategy

Answer 44

– 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

Question 45

Active Bond Management: Sources of Potential Profit (5)

Answer 45

1. Substitution swap – exchange of one bond for another more attractively priced bond with similar attributes 2. Intermarket spread swap – switching from one segment of the bond market to another (e.g., from Treasuries to corporates) 3. Rate anticipation swap – switch made between bonds of different durations in response to forecasts of interest rates 4. Pure yield pickup swap – moving to higher-yield, longer- term bonds to capture the liquidity premium 5. Tax swap – swapping two similar bonds to capture a tax benefit

Question 46

Horizon analysis

Answer 46

involves forecasting the realized compound yield over various holding periods of investment horizons