Valuing Bonds Flashcards
Bond certificate
states the terms of the bond
Maturity Date
final repayment date
Term
the time remaining until the repayment date
Coupon
promised interest payments
Face Value
notional amount used to compute the interest payments
Coupon Rate
determines the amount of each coupon payment, expressed as an APR
Coupon Payment
CPN = Coupon rate x Face Value
/ No. of coupon payments per year
Zero-coupon bond
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
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
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%
Yield to maturity of an n-year zero coupon bond
= (FV/price)^1/n - 1
Risk-Free Interest rates
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
Spot interest rate
another term for default-free, zero-coupon yield
zero-coupon yield curve
a plot of the yield of risk-free zero-coupon bonds as a function of the bonds maturity date
Coupon bonds
pay face value at maturity
pay regular coupon interest payments
Treasury notes
US treasury coupon security with original maturities of 1-10 years
Treasury bonds
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)
YTM of a coupon bond
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
Discount
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
Par
bond is at par if it equals the face value
Premium
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
Time and Bond Prices
- 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
Par value equation
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
Interest rate changes and bond prices
- 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
The sensitivity of a bonds price changes in interest rates is measured by the bonds duration
- bonds with high durations are highly sensitive to interest rate changes
- bonds with low durations are less sensitive to interest rate changes
Replicating a coupon bond
replicating a 3 year $1000 bond that pays 10% annual coupon using 3 zero-coupon bonds
will produce same payment when they have matured
Valuing a coupon bond using zero-coupon bond yields
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
Treasury Yield Curves
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
Corporate bond yields
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
No default
1-year, zero coupon $1000 treasury bill with YTM of 4%
P = 1000/1+YTM = 1000/1.04 = $961.54
Certain default
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
Risk of default
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
Bond ratings
AAA - best quality, small degree of risk, interest payments protected AA A BBB BB B/B CCC CC D - lowest rates, extremely poor prospects
Default spread
also known as credit spread
- difference between yield on corporate bonds and treasury yields
Sovereign bonds
- 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)
