1

"An appetising spread"

• Corporate bond spreads are now very high

• Between June 2007 and November 2008, junk issuers spread went from 2.5% to 15%

• This indicates default levels not seen since the depression

• This indicates that investors also care about the recovery rate. It is assumed to be 40%, but with the banking crisis some investors get less than 10 cents on the dollar

2

• Other reason: With the massive issuance of government debt, demand for corporate bonds is shrinking.

• Other reason: hedge funds have been forced to sell bonds to repay investors.

3

"The bond bubble" (Jan 8th)

• Investors retreat to safety, the demand for governement bond increases, mostly for US bonds. This pushes yields to historic lows. Is it a bond market bubble?

• US: 10-year treasuries were 2.6% in January 2009, down from 3.8% in January 2008

• Yields inUK: 3.3%, down from 4.5%.

• Less credit worthy governments benefit less from the low yields. Italian gvt bonds spread over the US Treasuries is 212 points, up from 124 points a year ago.

• Worst affected countries are Iceland, Ukraine, other eastern Europeans.

4

"Too much of a good thing" (Feb 5th)

• Has the bond bubble burst? Since december yields have risen from 2.04% to 2.9%.

• Reasons for this recent trend:

– The panicked selling of other securuties (equity, corporate bonds etc.) has stopped

– The thread of deflation has stopped, inducing investors to buy inflation protected bonds

– The stimulus package has increased the supply of gvt bonds

– Growing appetite for corporate bonds?

5

The term structure of interest rates

6

Spot rates

• The spot rate of a bond refers to rate for immediate settlement

• Example: Consider a bond that pays M in two years, with a price P. The spot rate r2 is such that:

22 )1( r

MP

7

The yield curve

The term structure of interest rates is a way of describing the

relationship between spot rates for bonds of different maturities.

r1 is the interest rate (in annual terms) on a 1-year bond

r2 is the interest rate (in annual terms) on a 2-year bond

r5 is the interest rate (in annual terms) on a 5-year bond

r10 is the interest rate (in annual terms) on a 10-year bond

8

Term structure of interest rates

00.010.020.030.040.05

0 2 4 6 8 10 12

Years to maturity

Yie

ld t

o m

atu

rity

9

Key questions

• How to interpret the yield curve?

• What does it tell about the market expectations?

• What does it tell about the default probability of corporate bonds?

• Link to investment banks: underwriting, portfolio management

10

Forward rates

• Suppose that the spot rate on a 1-year bond is 9%, and the 10% on a 2-year bond.

• What does this tell you about the future spot rates?

• As an investor you can choose between:1) Buy a 2-year bond with 10% YTM2) Buy a 1-year bond with 9% YTM, and at year 1 reinvest in another 1-year bond.

11

t=0 t=1 t=2

%91 r ?12 f

%102 r

12

• The price of the 2-year bond should be such that investors are indifferent between the two alternatives.

• If one alternative was less attractive to investors, its price would drop (yield go up), and the indifference would be restored.

13

• Forward rate: future spot rate implied by the bond prices

• Call the forward rate between year 1 and year 2.

• The “fair” forward rate is such that:

• Hence,

12f

11009.1)1(

)1(1

)1)(1()1(

1

22

12

1212

2

r

rf

frr

%1112 f

14

t=0 t=1 t=2

%91 r %1112 f

%102 r

15

With three periods

• Call the forward rate between year 2 and year 3.

• The “fair” forward rate is such that:

23f

22

33

23

232

23

3

)1(

)1(1

)1()1()1(

r

rf

frr

16

With n periods

• Call the forward rate between year m and year m+1.

• The “fair” forward rate is such that:

1, mmf

mm

mm

mm r

rf

)1(

)1(1

11

1,

17

Future contracts

Forward rates allow you to price future contracts.

Ex: At t=0 you agree to buy at t=1 a bond for £100 that pays

£111 at t=2. This is a future contract on a bond.

The forward price is £100 and is denoted . It is determined by the future rate and the payment

at maturity.

0F

12f

120 1 f

MF

18

Interpreting the yield curve: The pure expectations hypothesis

• In order to interpret the yield curve, we have to consider the investors preferences

• The PEH assumes that investors are risk neutral, and base their investment decisions only on expected returns

• Consider an investment of £A in a 3-year bond with yield

.

• The terminal value of investment is:

tr3

33 )1(£ trATV

19

• Consider the alternative investment of £A in a rolling 1-year bond for 3 years:

• No arbitrage requires that the investors are indifferent between the two alternatives:

))(1))((1)(1(£)( 21111 tttttt rErErATVE

))(1))((1)(1()1( 211113

3 tttttt rErErr

20

Informativity of the yield curve

If rates are expected to remain constant, then the yield curve

should be flat. Example:

If spot rates are expected to rise

%4

%4

32

2,11,1,1

tt

ttt

rr

rrr

%7%,6

%9%,8%,4

32

2,11,1,1

tt

ttttt

rr

rErEr

21

• Hence, the yield curve gives information about the market

expectations on future rates.

• Since interest rates are positively related to inflation and the state of the economy, an upward sloping yield curve could indicate that the market believes inflation will rise or the economic conditions will improve.

Yield Curve Slope Markets’ Guess of Where Rates are Headed

Flat No change in rates

Upward Rates will rise

Downward Rates will fall

23

Term structure of credit risk

Suppose we have the following yields:

Treasuries Corporate BBB

%12

%11

%10

12

2

1

f

r

r

%16

%14

%12

12

2

1

c

c

c

f

r

r

24

• Call the probability of repayment for the corporate

BBB bond. • Risk neutral investors would be indifferent

between the two bonds if:

• Hence the probability of default in year 1 is 1.8%.

01p

982.0)1(

)1()1()1(

1

1011101

c

c r

rprrp

25

• In reality, bondholders recover a fraction of the bond value when a company defaults. Let us denote the fraction of the bond value that is recovered.

• The credit spread, , is lower the higher is

)1()1()1()1( 1101101 rrprp cc

11 rrc

26

• Call the probability of no default between years 1 and 2.

• Hence, the marginal probability of default in year 2 is 3.5%

• The cumulative probability of default after 2 years is

12p

965.0)1(

1

1)1(

12

1212

121212

c

c

f

fp

ffp

%51 1201 pp

27

UK corporate bonds spreads (reference: government bonds

28

Probability of corporate bond default implied by the credit spreads