Lecture 9: Debt Markets and
Term Structure
Discount Bonds
• No coupon payments, just principal at
maturity date (conventionally, $100).
• Initially sold at a discount (less than $100)
and price rises through time, creating
income.
• Term T, Yield to Maturity (YTM) r
1
Pt 
(1  r )T
1
Pt 
(1  r / 2) 2T
Compound Interest
• If annual rate is r, compounding once per
year, balance = (1+r)t after t years.
• If compounded twice per year, balance is
(1+r/2)2t after t years.
• If compounded n times per year, balance is
(1+r/n)nt after t years.
• Continuous compounding, balance is ert.
Price & Yield on T-Bills
• For buyer, Price = 100-Discount
• Discount = asked*(Days to Maturity/360).
• Yield = (Discount/Price)(365/(Days to
Maturity)). (Unless maturity > 6 months, in
which case quadratic formula using semiannual compounding is required.)
Example Dec 18, 2000
• T-Bill maturing March 15, Asked=5.83%,
87 days to maturity.
• Discount = 5.83*87/360=1.40891
• Price=100-1.40891=98.59108
• Yield=(1.40891/98.59108)(365/87)=5.995%
Conventional Bonds Carry
Coupons
• Conventional Bond Issued at par (100),
coupons every six months.
• Term is time to maturity.
Pt 
1
1 1
100
Pt  c( 
)
T
r (1  r ) r (1  r ) T
c 1
1
1
100
Pt  (

)
2T
2 r / 2 (1  r / 2) r / 2 (1  r / 2) 2 T
Bond Yield Tables
Term Structure of Interest Rates
•
•
•
•
•
Yield to maturity plotted against term
Also called “The Yield curve”
Usually upward sloping
Inverted yield curve
Hump shaped yield curve
Term Structure of Interest Rates, 1999
and 2004
7
Nov-00
6
5
Yield
4
3
Jan-04
2
1
0
0.1
1
10
Maturity in Years
100
Causes of Interest Rates
• Eugen von Böhm-Bawerk: Capital and
Interest, 1884: technological progress, time
preferences, advantages to roundaboutness
• Irving Fisher 1867-1947, wrote Theory of
Interest 1930
Irving Fisher Yale ‘88
Irving Fisher Diary at Yale
• July 31, 1885 “it is neither politic nor right to
study at the expense of one’s health.” Rowing.
• “When I fall in love she must be a girl of pure
morality, broad culture and fine tastes.”
• “I have an earnest desire to be good and useful”
• April 4, 1886, roommate dies of a “cold.”
• May 29, 1887, “I take great satisfaction in my
election to Bones for I felt it to be my first little
conquest among men. As a freshman I was afraid
of my own voice.”
Irving Fisher Diagram Today
Forward Rates
• Forward rates are interest rates that can be
taken in advance using term structure
• J. R. Hicks Value and Capital 1939
(1  r2 ) 2  (1  r1 )(1  f 2 )
(1  rk ) k  (1  rk 1 ) k 1 (1  f k )
Example of Forward Rates
• Suppose I in 1925 expect to have £100 to invest in
1926, but want the money back by 1927. How can
I guarantee the interest rate on the £100
investment today (1925)?
• Buy in 1925 (1+r2 )2/(1+r1) 2-period discount
bonds maturing at £100 in 1927. Cost: £1/(1+r1)
• Short in 1925 one 1-period discount bond
maturing at £100 in 1926. Receive: £1/(1+r1)
• I have now locked in the interest rate 1+f=(1+r2)2/
(1+r1) between 1926 and 1927.
Expectations Theory
• Forward rates equal expected spot rates
• Slope of term structure indicates expected
future change in interest rates.
Liquidity Preference Hypothesis
• Forward rates equal expected future spot
rates plus a “risk premium.” (J. R. Hicks,
1939)
• Modigliani and Sutch: Risk premium could
be either positive or negative. Preferred
habitat hypothesis
Inflation and Interest Rates
• Nominal rate quoted in dollars, real rate
quoted market baskets
• Nominal rate usually greater than real rate.
(1  rmoney )  (1  rreal )(1  i )
rmoney  rreal  i
Indexed Bonds
• Paul Revere, Massachusetts, 1780
• U. S. Treasury, 1997
• TIPS Treasury Inflation Protection
Securities, $115 billion outstanding 2000,
2% of US national debt
• UK Index-Linked Gilts 20% of debt
• France recently issued Euro Index bonds