Valuing Bonds

Question 1

Bond certificate

Answer 1

states the terms of the bond

Question 2

Maturity Date

Answer 2

final repayment date

Question 3

Term

Answer 3

the time remaining until the repayment date

Question 4

Coupon

Answer 4

promised interest payments

Question 5

Face Value

Answer 5

notional amount used to compute the interest payments

Question 6

Coupon Rate

Answer 6

determines the amount of each coupon payment, expressed as an APR

Question 7

Coupon Payment

Answer 7

CPN = Coupon rate x Face Value

/ No. of coupon payments per year

Question 8

Zero-coupon bond

Answer 8

doesn’t make coupon payments

always sells at a discount (below face value) so they are also called pure discount bonds

• Treasury Bills are US gov zero-coupon bonds with a maturity of up to a year
• compensation for zero-coupon bonds is the difference between the initial price and the face value

Question 9

Yield to maturity

Answer 9

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

Price of zero coupon bonds:

P = FV/(1+YTM)^n

e.g for $100,000 zero coupon bond with with 1 year

96,618.36 = 100,000/(1+YTM)^!

YTM = 3.5%

Question 10

Yield to maturity of an n-year zero coupon bond

Answer 10

= (FV/price)^1/n - 1

Question 11

Risk-Free Interest rates

Answer 11

a default-free zero-coupon 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 bond

risk free interest rate with maturity, n:

r = YTMn

Question 12

Spot interest rate

Answer 12

another term for default-free, zero-coupon yield

Question 13

zero-coupon yield curve

Answer 13

a plot of the yield of risk-free zero-coupon bonds as a function of the bonds maturity date

Question 14

Coupon bonds

Answer 14

pay face value at maturity

pay regular coupon interest payments

Question 15

Treasury notes

Answer 15

US treasury coupon security with original maturities of 1-10 years

Question 16

Treasury bonds

Answer 16

US treasury coupon security with original maturities over 10 years

e.g US treasury issued a 5-year, $1000 bond with 5% coupon rate and semi annual coupons. What’re the cash flows until maturity

CPN = $1000 x 5%/2 = $25

so receive 25 dollars every 6 months and $1000 when bond matures (after 10 six-month periods)

Question 17

YTM of a coupon bond

Answer 17

the single discount rate that equates the present value of the bonds remaining cash flows to its current price

P = CPN/Y x (1-(1/(1+Y)^N))+ FV/(1+Y)^N

where FV = face value

e. g that 5-year $1000 bond has 6.3% APR compounded semi annually (CPN = 25 dollar payments every 6 months)
6. 3/3 = 3.15%

P = 25/0.0315 x (1-1/1.0315^10) + 1000/1.0315^10

Question 18

Discount

Answer 18

bond is at a discount if price is less than the face value

if a coupon bond trades at a discount, an investor will earn a return both from receiving the coupon and from receiving a face value that exceeds the price paid for the bond

its yield to maturity will exceed the coupon rate

Question 19

Par

Answer 19

bond is at par if it equals the face value

Question 20

Premium

Answer 20

bond is selling at a premium if price is greater than face value

• if a bond trades at a premium, it 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
• its yield to maturity will be smaller than its coupon rate
  e. g 30 year bond with annual coupon payment with a 10% coupon rate. Yield to maturity is 5%, what is the price of each bond per $100 face value?

P = 10/0.05 x (1-1/1.05^30) +100/1.05^30 = $176.86

so bond selling at a premium

Question 21

Time and Bond Prices

Answer 21

• holding all other things constant, a bonds yield to maturity will not change over time
• Yield to maturity is the investors required return to hold this bond

this takes into account risk-free rate and risk premium

if they stay the same, YTM won’t change

• holding all other things constant, the price of discount or premium bond will move to par value over time

Question 22

Par value equation

Answer 22

P = CPN/Y x (1-1/(1+Y)^N) + FV/(1+Y)^N

then the date is getting nearer to maturity rate there are:

• less coupon payments left
• less discount of face value
• so will converge towards face value

Question 23

Interest rate changes and bond prices

Answer 23

• there’s an inverse relationship between interest rates and bond prices

as interest rates and bond prices rise, bond prices fall and vice versa

everything else equal:

• shorter-maturity zero-coupon bonds are less sensitive to interest rate changes than longer-maturity ones
• bonds with higher coupon rates are less sensitive to interest rate changes team bonds with lower coupon rates

Question 24

The sensitivity of a bonds price changes in interest rates is measured by the bonds duration

Answer 24

• bonds with high durations are highly sensitive to interest rate changes
• bonds with low durations are less sensitive to interest rate changes

Question 25

Replicating a coupon bond

Answer 25

replicating a 3 year $1000 bond that pays 10% annual coupon using 3 zero-coupon bonds

will produce same payment when they have matured

Question 26

Valuing a coupon bond using zero-coupon bond yields

Answer 26

the price of a coupon bond must equal the present values of its coupon payments and its face value

PV = CPN/1+YTM1 + CPN/(1+YTM2)^2 etc + (CPN +FV)/(1+YTMn)^n

where CPN is each coupon payment
FV = face value

how to value a coupon bond using zero-coupon bond yields

Question 27

Treasury Yield Curves

Answer 27

Treasury coupon-paying yield curve

• on the run bonds are the most recently issued bonds ]

the yield curve for these often a plot of the yield on these bonds

Question 28

Corporate bond yields

Answer 28

Investors pay less for bonds with credit risk than they would for an identical default-free bond

• the yield of bonds with credit risk will be higher than that of otherwise identical default-free bonds

Question 29

No default

Answer 29

1-year, zero coupon $1000 treasury bill with YTM of 4%

P = 1000/1+YTM = 1000/1.04 = $961.54

Question 30

Certain default

Answer 30

suppose now bond issuer will pay 90% of the obligation

P = 900/1+YTM = 900/1.04 = $865.38

when computing YTM of certain default bonds, promised rather than actual cash flows are used

• the YTM of a certain default bond is not equal to the expected return of investing in the bond. The YTM will always be higher than the expected return of investing in the bond

Question 31

Risk of default

Answer 31

consider a one year $1000 zero-coupon bond

• there’s a 50% chance the bond will repay its full-value, 50% chance it’ll default and you’ll receive $900. Thus you would expect to receive $950
• because of the uncertainty, discount rate is 5.1%

P = 950/1.051 = $903.90

YTM = FV/P -1 = 1000/903.90 -1 = 0.1063

• a bonds expected rerun will be less than the YTM if there’s a risk of default
• a higher YTM doesn’t necessarily imply that a bonds expected return is higher

Question 32

Bond ratings

Answer 32

AAA - best quality, small degree of risk, interest payments protected
AA
A
BBB
BB
B/B
CCC
CC
D - lowest rates, extremely poor prospects

Question 33

Default spread

Answer 33

also known as credit spread

• difference between yield on corporate bonds and treasury yields

Question 34

Sovereign bonds

Answer 34

• bonds issued by national governments
  e. g US treasury securities, usually default free
• All sovereign bonds are not default free
• important of inflation expectations (potential to inflate away the debt)