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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
Tt rP
)1(
1
Tt rP
2)2/1(
1
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 semi-annual 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.
P cr r r rt T T
(
( ))
( )
1 1
1
1 1 0 0
1
tP
Pc
r r r rt T T
2
1
2
1
1 2
1
2
1 0 0
1 22 2(/ ( / ) /
)( / )
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
0
1
2
3
4
5
6
7
0.1 1 10 100
Maturity in Years
Yie
ld
Nov-00
Jan-04
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)(1()1( 212
2 frr
)1()1()1( 11 k
kk
kk frr
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)(1()1( irr realmoney
irr realmoney
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
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