Yield Curve and Term
Structure of Interest Rate
• Base rate of interest
– US Treasuries are “safer” than any other (US)
debt security
• free of credit risk
• free of liquidity risk
– On-the-Run Treasuries (most recently auctioned) more
so than Off-the-Run Treasuries
• free of call risk
• but NOT free of interest rate risk
– So, their yield is the minimum rate that a (US)
investor would accept
• This rate is called the base or benchmark rate
Other rates trade at a spread (or risk premium) to
the benchmark
• Yield curve
– “Plot of yield vs maturity”
• Note: Bonds of same maturity with different coupon
trade at different yields
• Yield curve typically is a par yield curve (same as
coupon yield curve).
– For Treasuries, use mainly on-the-run issues (whose
coupon issues typically trade close to par)
• Different shapes of Treasury yield curve
– yield increasing with maturity: “upward sloping” or “normal”
– Yield decreasing with maturity: “downward sloping” or
“inverted”
– Hump / U-shaped structure: max/min yield at intermediate
maturities
• Treasury yield curve useful in setting
benchmark for other bonds, but…
– Yield curve is not very precise in pricing a
premium or discount bond
– Yield assumes a single discount rate for all
cash flows
• If coupons differ, cash flow patterns differ
• not appropriate to use same yield to discount
different cash flows even with the same maturity
• The building blocks of treasury security is
zero coupon bonds, or zeros
– Also called pure discount bonds
– The yield implied by a zero is called spot rate
of interest
– The price (PV) per dollar of par value for a
zero coupon bond is called the discount factor
• Other coupon bond, or any risk-free cash
flows can be thought as a portfolio of zero
coupon bonds
• No-arbitrage condition says the price of the
portfolio has to be the same as the sum of
its component assets
An Example
Bond A is a one period zero traded at 3% yield
Bond B is a two-period zero traded at 4% yield
Bond C is a two-period coupon bond that pays 10%
coupon.
Bond D is a two-period coupon bond that pays 20%
coupon.
Q: What is the price of bond C and D?
Q: What is the ytm of bond C and D?
Q: What coupon rate needed to be offered for a
par bond?
Q: What should be the ranking of all bonds in
terms of yield to maturity?
• In practice, intermediate to long term
treasury securities are typically issued with
coupons.
• Strictly speaking, term structure of interest
rate and yield curve should be the yields for
zeros to avoid ambiguity
• We can reverse engineer the zero coupon
bond yield with coupon bonds
– The process is called bond bootstrapping
• Sequential substitution
– Q: How do we find one and two year spot rates
if we only know the price of bond A and bond C?
Q: How do we find one and two year spot rates if
we only know the price of bond A and bond C?
Forward Rates
• Embedded in spot rate curve is fact that future spot rates
can be “locked in” today.
– This does not mean that future spot rates will actually be this
“lockable” rate
– But this does represent investor expectations
• Risk premium
– And I can arrange to borrow/lend tomorrow at this rate (which is
known today)
• This known future spot rate is called the forward rate,
fn(t)
– E.g., f5(5) = interest rate you can lock in applicable at year 5 for
a deposit for 5 periods (matures at period 10)
– Q: How is forward rate related to spot rates?