Valuing Debt

FINA 7330Corporate Finance

Lecture 13

Topics Covered

• Real and Nominal Rates of Interest

• Term Structure and Yield to Maturity

• The Term Structure and Bond Pricing

• Theories of the Term Structure

Irving Fisher and the Theory of Interest Rates

• The Real Interest Rate is determined by the real economic activity and demographics of the Economy (The Demand and Supply for Capital)

• The Nominal Interest Rate is the real rate adjusted for inflation

1 + R = (1 + r) * (1 + E[inf.])

U.K. Rates

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Nominal Yield on UK 10 yr bonds

Real Yield on UK 10 yr bonds

Inflation and Interest Rates

• Note the Real Rate tends to be rather stable but the Nominal Rate is more volatile. What makes it more volatile? The Expected inflation rate!

• Does the theory fit the facts? We can’t measure Expected inflation, but assume actual inflation follows expected inflation closely then:

The Return on US Treasury Bills and the Inflation rate (1953-2003)

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Treasury BillsInflation

Normal Yield Curve

Current Yield Curve

Treasury Yields

• Maturity Yield

• 12/13/2006 5.17• 12/13/2007 5.03• 12/12/2008 4.76• 12/12/2009 4.66• 12/12/2011 4.60• 12/11/2013 4.60• 12/10/2016 4.61• 12/8/2026 4.80• 12/5/2036 4.71

How to Determine Yield Curve

• WSJ U.S. Treasury Strips Maturity Type Yield Nov 06 ci 4.51Nov 06 np 4.72Feb 07 bp 4.96May 07 np 4.95Aug 07 np 4.92…………………………………….May 16 bp 4.66Aug 16 bp 4.61Nov 16 bp 4.69

The Term Structure and Bond Prices

Consider Two Year Treasury (3.125s October 08) Price is 97:03 = 97.09

4/07 10/07 4/08 10/08 1.5625 1.5625 1.5625 101.5625

YTM:

The Term Structure and Bond Prices

Consider Two Year Treasury (3.125s October 08) Price is 97:03

4/07 10/07 4/08 10/08 1.5625 1.5625 1.5625 101.5625

YTM: 4.66

But consider the term structure Two Year Treasury

(3.125s October 08) Price is 97:03

4/07 10/07 4/08 10/08 1.5625 1.5625 1.5625 101.5625Strip Yields

4.94 4.87 4.50 4.64PV 1.5248 1.4899 1.4616 92.7551Value = 97.23

Bond Prices and Yields

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5 Year 9% Bond 1 Year 9% Bond

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Duration Calculation

Year Ct PV(Ct) at 2.75%Proportion of Total

Value [PV(Ct)/V]Proportion of Total

Value Time

1 55 53.53 0.049 0.04492 55 52.1 0.047 0.0943 55 50.7 0.046 0.1384 1055 946.51 0.858 3.433

V = 1102.83 1 Duration= 3.714 years

Risk and Duration

• The relationship between Risk and Duration

• Volatility = Duration/(1 + YTM)

• So in example, D = 3.714, YTM = 2.75%

Volatility = 3.615% % ChangeValue at 3.25% 1083.14 -1.78Value at 2.25% 1123.04 1.83Volatiltiy 3.61%

Duration

Year CF PV@YTM % of Total PV % x Year

1 68.75 65.54 .060 0.060

2 68.75 62.48 .058 0.115

3 68.75 59.56 .055 0.165

4 68.75 56.78 .052 0.209

5 1068.75 841.39 .775 3.875

1085.74 1.00 Duration 4.424

Example (Bond 1)Calculate the duration of our 6 7/8 % bond @ 4.9 % YTM

Example

rn is the “Spot Rate” = the annualized yield on a discount bond making 1 payment n years in the future

fn is the “Forward Rate” = the implied yield on a one year discount bond issued n-1 years in the future.

Spot/Forward rates

Spot and Forward Rates

• In general:

• (1+ rn) = (1 + r1)(1 + f2)(1 + f3)…)(1 + fn)

ExampleWhat is the 3rd year forward rate?2 year zero treasury YTM = 4.633 year zero treasury YTM = 4.57

Spot/Forward rates

• ExampleWhat is the 3rd year forward rate?2 year zero treasury YTM = 4.633 year zero treasury YTM = 4.57

Answer(1+r3)3 = (1+r2)2(1+f3)

(1+r3)3/(1+r2)2 = (1 + f3)

1.1435/1.0947 = 1.0445

f3 = 4.45%

Spot/Forward rates

Matrix Pricing