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

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Tt rP

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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.

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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

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1

2

3

4

5

6

7

0.1 1 10 100

Maturity in Years

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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

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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.

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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