Fi8000 Valuation of Financial Assets pot

Isabel TkatchTkatch Assistant Professor of Finance Debt instruments ☺ ☺Types of bonds Types of bonds ☺ ☺Ratings of bonds default risk Ratings of bonds default risk ☺ ☺Spot and forward in

Fi8000 Valuation of

Financial Assets

Fall Semester 2009

Dr Isabel

Dr Isabel TkatchTkatch Assistant Professor of Finance

Debt instruments

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☺Types of bonds Types of bonds

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☺Ratings of bonds (default risk) Ratings of bonds (default risk)

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☺Spot and forward interest rate Spot and forward interest rate

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☺Spot and forward interest rate Spot and forward interest rate

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☺The yield curve The yield curve

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☺Duration Duration

Bond Characteristics

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☺A bond is a security issued to the lender (buyer) by the A bond is a security issued to the lender (buyer) by the

borrower (seller) for some amount of cash

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☺The bond obligates the issuer to make specified The bond obligates the issuer to make specified

payments of interest and principal to the lender, on

specified dates

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☺The typical The typical coupon bond coupon bond obligates the issuer to make obligates the issuer to make

coupon payments, which are determined by the

coupon payments, which are determined by the coupon coupon

rate

rate as a percentage of the as a percentage of the par value par value ((face value face value) )

When the bond matures, the issuer repays the par value

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☺Zero Zero coupon bonds coupon bonds are issued at discount (sold for a are issued at discount (sold for a

price below par value), make no coupon payments and

pay the par value at the maturity date

Bond Pricing Bond Pricing Examples Examples

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☺The par value of a riskThe par value of a risk free zero coupon bond is free zero coupon bond is

$100 If the continuously compounded risk

$100 If the continuously compounded risk free free rate is 4% per annum and the bond matures in three months, what is the price of the bond today?

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☺A risky bond with par value of $1,000 has an A risky bond with par value of $1,000 has an annual coupon rate of 8% with semiannual installments If the bond matures 10 year from now and the risk

now and the risk adjusted cost of capital is 10% adjusted cost of capital is 10%

per annum compounded semiannually, what is the price of the bond today?

Yield to Maturity

Yield to Maturity Examples Examples

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☺What is the What is the yield to maturity yield to maturity (annual, (annual,

compounded semiannually) of the risky coupon

compounded semiannually) of the risky

coupon bond, if it is selling at $1,200?

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☺What is the What is the expected yield to maturity expected yield to maturity of the p p y y y yof the

risky coupon

risky coupon bond, if we are certain that the bond, if we are certain that the

issuer is able to make all coupon payments but

we are uncertain about his ability to pay the par

value We believe that he will pay it all with

probability 0.6, pay only $800 with probability

0.35 and won’t be able to pay at all with

probability 0.05

Default Risk and Bond Rating

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☺Although bonds generally promise a fixed flow of Although bonds generally promise a fixed flow of income, in most cases this cash

income, in most cases this cash flow stream is uncertain flow stream is uncertain since the issuer may default on his obligation

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☺US government bonds are usually treated as free of US government bonds are usually treated as free of default (credit) risk Corporate and municipal bonds are default (credit) risk Corporate and municipal bonds are considered risky

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☺Providers of bond quality rating:Providers of bond quality rating:

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☺Moody’s Investor Services Moody’s Investor Services

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☺Standard and Poor’s Corporation Standard and Poor’s Corporation

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☺Duff & Phelps Duff & Phelps

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☺Fitch Investor Service Fitch Investor Service

Default Risk and Bond Rating

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☺AAA (Aaa) is the top rating.AAA (Aaa) is the top rating

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☺Bonds rated BBB (Baa) and above are Bonds rated BBB (Baa) and above are

considered

considered investment investment grade bonds grade bonds

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☺Bonds rated lower than BBB are considered Bonds rated lower than BBB are considered

l ti d j j k b k b d d

speculative

speculative grade grade or or junk bonds junk bonds

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☺Risky bonds offer a riskRisky bonds offer a risk premium The greater premium The greater

the default risk the higher the

the default risk the higher the default risk default

risk premium

premium

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☺The The yield spread yield spread is the difference between the is the difference between the

yield to maturity of high and lower grade bond

Estimation of Default Risk

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☺The determinants of the The determinants of the bond default risk bond default risk (the (the probability of bankruptcy) and

probability of bankruptcy) and debt quality ratings debt quality ratings

are based on measures of financial stability:

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☺Ratios of earnings to fixed costs; Ratios of earnings to fixed costs;

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☺Leverage ratios; Leverage ratios; g g ; ;

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☺Liquidity ratios; Liquidity ratios;

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☺Profitability measures; Profitability measures;

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☺Cash Cash flow to debt ratios flow to debt ratios.

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☺A complimentary measure is the A complimentary measure is the transition matrix transition matrix ––

estimates the probability of a change in the rating of the bond

The Term

The Term Structure of Interest Rates Structure of Interest Rates

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☺ The The short interest rate short interest rate is the interest rate is the interest rate

for a given time interval (say one year,

which does not have to start today).

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☺ The The yield to maturity yield to maturity ((spot rate spot rate) is the ) is the

internal rate of return (say annual) of a zero

coupon bond, that prevails today and

corresponds to the maturity of the bond.

Example

In our previous calculations we’ve assumed that all the

that all the short interest rates short interest rates are equal Let are equal Let

us assume the following:

Interest rate

For the time interval

Example

What is the price of the 1, 2, 3 and 4 years

zero

zero coupon bonds paying $1,000 at coupon bonds paying $1,000 at

maturity?

Maturity ZeroZero Coupon Bond PriceCoupon Bond Price

22 $1,000/(1.08*1.10) = $841.75

33 $1,000/(1.08*1.10*1.11) = $758.33

44 $1,000/(1.08*1.10*1.1122) = $683.18

Example

What is the yield What is the yield to to maturity of the 1, 2, 3 and maturity of the 1, 2, 3 and

4 years zero

4 years zero coupon bonds paying $1,000 at coupon bonds paying $1,000 at maturity?

The Term

The Term Structure of Interest Rates Structure of Interest Rates

The price of the zero

The price of the zero coupon bond is calculated coupon bond is calculated

using the

using the short interest rates short interest rates (r(rtt, t = 1,2…,T) For

a bond that matures in T years there may be up to

T different short rates

Price = FV / [(1+r11)(1+r22)…(1+rT)]

The

The yield yield to to maturity maturity (y(yT) of the zero) of the zero coupon coupon

bond that matures in T years, is the internal rate of

return of the bond cash flow stream

Price = FV / (1+yT))T

The Term The Term Structure of Interest Rates Structure of Interest Rates

The price of the zero The price of the zero coupon bond paying $1,000 coupon bond paying $1,000

in 3 years is calculated using the short term rates:

Price = $1,000 / [1.08*1.10*1.11] = Price = $1,000 / [1.08*1.10*1.11] = $758.33 $758.33 The

The yield yield to to maturity maturity (y(y33) of the zero) of the zero coupon coupon bond that matures in 3 years solves the equation

$758.33

$758.33 = $1,000 / (1+y = $1,000 / (1+y33))33

yy33= = 9.660%. 9.660%.

The Term

The Term Structure of Interest Rates Structure of Interest Rates

Thus the

Thus the yields yields are in fact are in fact geometric geometric

averages

averages of the of the short interest rates short interest rates in in

each period

(1+y

(1+y ))T= (1+r = (1+r )(1+r )(1+r ) (1+r ) (1+r ))

(1+yT))T= (1+r11)(1+r22)…(1+rT))

(1+yT) = [(1+r11)(1+r22)…(1+rT)](1/T)

The

The yield curve yield curve is a graph of bond yield is a graph of bond yield to

to maturity as a function of time

maturity as a function of time to to maturity maturity.

The Yield Curve (Example)

YTM

9.660%

Time to Maturity

8.000%

8.995%

The Term

The Term Structure of Interest Rates Structure of Interest Rates

If we assume that all the

If we assume that all the short interest rates short interest rates ((rrtt, t

= 1, 2…,T) are equal, then all the

= 1, 2…,T) are equal, then all the yields yields ((yyT) of

zero

zero coupon bonds with different maturities (T = 1, coupon bonds with different maturities (T = 1,

2…) are also equal and the yield curve is flat

A

A flat flat yield curve is associated with an expectedyield curve is associated with an expected

A

A flat flat yield curve is associated with an expected yield curve is associated with an expected

constant interest rates in the future;

An

An upward sloping upward sloping yield curve is associated with yield curve is associated with

an expected increase in the future interest rates;

A

A downward sloping downward sloping yield curve is associated yield curve is associated

with an expected decrease in the future interest

rates

The Forward Interest Rate

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☺ The The yield to maturity yield to maturity ((spot rate spot rate) is the internal ) is the internal rate of return of a zero coupon bond, that prevails today and corresponds to the maturity

of the bond

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☺ TheThe forward interest rate forward interest rate is the rate of return ais the rate of return a

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☺ The The forward interest rate forward interest rate is the rate of return a is the rate of return a borrower will pay the lender, for a specific loan, taken at a specific date in the future, for a specific time period If the principal and the interest are paid at the end of the period, this loan is equivalent to a forward zero coupon bond

The Forward Interest Rate

Suppose the price of 1

Suppose the price of 1 year maturity zeroyear maturity zero coupon coupon

bond with face value $1,000 is $925.93, and the

price of the 2

price of the 2 year zeroyear zero coupon bond with $1,000 coupon bond with $1,000

face value is $841.68

If there is no opportunity to make arbitrage profits,

what is the 1

what is the 1 year forward interest rate for the year forward interest rate for the

second year?

How will you construct a synthetic 1

How will you construct a synthetic 1 year forward year forward

zero

zero coupon bond (loan of $1,000) that coupon bond (loan of $1,000) that

commences at t = 1 and matures at t = 2?

The Forward Interest Rate

If there is no opportunity to make arbitrage profits, the 1

the 1 year forward interest rate for the second year year forward interest rate for the second year must be the solution of the following equation:

(1+y (1+y ))22= (1+y = (1+y )(1+f )(1+f )) (1+y22))22= (1+y11)(1+f22), where

yyT= yield to maturity of a T= yield to maturity of a T year zeroyear zero coupon bondcoupon bond

fftt= 11 year forward rate for year tyear forward rate for year t

The Forward Interest Rate

In our example, yy11= 8% and y22= 9% Thus,

(1+0.09)22= (1+0.08)(1+f22))

ff = 0 1001 = 10 01% = 0 1001 = 10 01%

ff22= 0.1001 = 10.01%.

Constructing the loan (borrowing):

1 Time t = 0 CF should be zero;

2 Time t = 1 CF should be +$1,000;

3 Time t = 2 CF should be

3 Time t = 2 CF should be $1,000(1+f$1,000(1+f22) = ) = $1,100.1.$1,100.1

The Forward Interest Rate

Constructing the loan:

we would like to borrow $1,000 a year from now for a forward interest rate of 10.01%

1

1 (#3) CF(#3) CF( ) 0000= $925.93 but it should be zero We offset that cash flow if we buy the 1

cash flow if we buy the 1 year zero coupon bond for year zero coupon bond for

$925.93 That is, if we buy $925.93/$925.93 = 1 units of the 1

the 1 year zero coupon bond;year zero coupon bond;

2

2 (#1) CF(#1) CF11should be equal to $1,000;

3

3 (#2) CF(#2) CF22= = $,1000*1.1001 = $,1000*1.1001 = $1,100.1 We generate that $1,100.1 We generate that cash flow if we sell 1.1001 of the 2

cash flow if we sell 1.1001 of the 2 year zeroyear zero coupon coupon bond for 1.1001* $841.68 = $925.93

Bond Price Sensitivity

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☺ Bond prices and yields are inversely related.Bond prices and yields are inversely related

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☺ Prices of Prices of long long term bonds term bonds tend to be more tend to be more

sensitive to changes in the interest rate

(required rate of return / cost of capital) than

those of short

those of short term bonds (compare two zeroterm bonds (compare two zero

those of short

those of short term bonds (compare two zero term bonds (compare two zero

coupon bonds with different maturities)

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☺ Prices of Prices of high coupon high coupon rate rate bonds bonds are less are less

sensitive to changes in interest rates than

prices of low coupon

prices of low coupon rate bonds (compare a rate bonds (compare a

zero

zero coupon bond and a couponcoupon bond and a coupon paying bond paying bond

of the same maturity)

Duration

The observed bond price properties suggest that the

timing timing and and magnitude magnitude of of all cash flows all cash flows affect bond affect bond prices, not only time

prices, not only time toto maturity maturity Macaulay’s duration Macaulay’s duration is a is a measure that summarizes the timing and magnitude effects

of all promised cash flows.pp

1

Cash flow weight:

/ 1

Macauley's Duration:

t t t

T t t

w BondPrice

D =t w

+

=

Calculate the duration of the following bonds:

1.

1 8% coupon bond; $1,000 par value; 8% coupon bond; $1,000 par value;

semiannual installments; Two years to

maturity; The annual discount rate is

maturity; The annual discount rate is

10%, compounded semi

10%, compounded semi annually annually.

2.

2 Zero Zero coupon bond; $1,000 par value; coupon bond; $1,000 par value;

Two year to maturity; The annual

discount rate is 10%, compounded semi

discount rate is 10%, compounded

semi annually.

Properties of the Duration

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☺ The The duration of a zero duration of a zero coupon bond coupon bond

equals its time to maturity;

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☺ Holding maturity and par value constant, Holding maturity and par value constant, the bond’s

the bond’s duration is lower duration is lower when the when the the bond s

the bond s duration is lower duration is lower when the when the

coupon rate is higher coupon rate is higher;;

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☺ Holding coupon Holding coupon rate and par value rate and par value constant, the bond’s

constant, the bond’s duration generally duration generally increases

increases with its with its time to maturity time to maturity

Macaulay’s Duration

Bond price (p) changes as the bond’s yield

to maturity (y) changes We can show that

the proportional price change is equal to the

proportional change in the yield times the

proportional change in the yield times the

duration.

(1 ) (1 )

D

= − ⋅

+

Modified Duration

Practitioners commonly use the modified duration measure

duration measure D*=D/(1+y), D*=D/(1+y), which can be which can be presented as a measure of the bond price sensitivity to changes in the interest rate sensitivity to changes in the interest rate

*

P

P

Δ = − ⋅ Δ

Example

Calculate the percentage price change for the following

bonds, if the semi

bonds, if the semi annual interest rate increases from 5% to annual interest rate increases from 5% to

5.01%:

1 8% coupon bond; $1,000 par value; semiannual

installments; Two years to maturity; The annual

discount rate is 10%, compounded semi

discount rate is 10%, compounded semi annually.annually

d scou a e s 0%, co pou ded se

d scou a e s 0%, co pou ded se a ua y

2 Zero coupon bond; $1,000 par value; Two year to Zerocoupon bond; $1,000 par value; Two year to

maturity; The annual discount rate is 10%,

compounded semi

compounded semi annually.annually

3 A zero coupon bond with the same duration as the 8% A zerocoupon bond with the same duration as the 8%

coupon bond (1.8852 years or 3.7704 6

coupon bond (1.8852 years or 3.7704 6 months months

periods The modified duration is 3.7704/1.05 = 3.591

66 months periods).months periods)

Example

The percentage price change for the following bonds as a result of an increase in the interest rate (from 5% to 5.01%):

1 ∆P/P = ∆P/P = D*·∆y = D*·∆y = (3.7704/1.05)·0.01% = (3.7704/1.05)·0.01% = 0.03591%0.03591%

2 ∆P/P = ∆P/P = D*·∆y = D*·∆y = (4 /1.05)·0.01% = (4 /1.05)·0.01% = 0.03810%0.03810%

3 ∆P/P = ∆P/P = D*·∆y = D*·∆y = (3.7704/1.05)·0.01% = (3.7704/1.05)·0.01% = 0.03591%0.03591%

Note that:

When two bonds have the same duration (not time to maturity) they also have the same price sensitivity to changes in the interest rate: 1 vs 3

When the duration (not time When the duration (not time toto maturity) is higher for maturity) is higher for one of the bonds then the price sensitivity of that bond

is also high: 1 vs 2; 3 vs 2

The Use of Duration

☺

☺ It is a simple It is a simple summary statistic summary statistic of the effective of the effective

average maturity of the bond (or portfolio of

fixed income instruments);

☺

☺ Duration can be presented as aDuration can be presented as a measure of measure of

☺

☺ Duration can be presented as a Duration can be presented as a measure of measure of

bond (portfolio) price sensitivity

bond (portfolio) price sensitivity to changes to changes

in the interest rate (cost of capital);

☺

☺ Duration is an essential Duration is an essential tool in portfolio tool in portfolio

immunization:

immunization: hedging interest rate risk.hedging interest rate risk

Uses of Interest Rate Hedges

☺

☺ Owners of fixedOwners of fixed income portfolios protecting income portfolios protecting against a rise in rates

☺

☺ Corporations planning to issue debt securities Corporations planning to issue debt securities protecting against a rise in rates

☺

☺ Investor hedging against a decline in rates for a Investor hedging against a decline in rates for a planned future investment

☺

☺ Exposure for a fixedExposure for a fixed income portfolio is income portfolio is proportional to modified duration

Hedging Interest Rate Risk: Textbook p 802

Portfolio value = $10 million

Modified duration = 9 years

If rates rise by 10 basis points (

If rates rise by 10 basis points (bp bp) ) ΔΔy = y = ( 1% ) ( 1% )

Change in value = D*·∆y = ( 9 ) ( 1% ) = ( 9% ) or $90,000

Price value of a basis point (PVBP) =

$90,000 / 10

$90,000 / 10 bp bp = $9,000 = $9,000 PVBP:

PVBP: measures dollar value sensitivity to changes in measures dollar value sensitivity to changes in

interest rates

Hedging Interest Rate Risk: Text Example

Hedging strategy: offsetting position in Treasury bonds futures

T T Bond futures contract calls for delivery of $100,000 par Bond futures contract calls for delivery of $100,000 par value T

value T Bonds with 6% coupons and 20 Bonds with 6% coupons and 20 years maturity years maturity.

Assumptions Assumptions::

Contract Modified duration = D* = 10 years Futures price = F 00 = $90 per $100 par value

(i.e., contract multiplier = 1,000)

Hedging Interest Rate Risk: Text Example

If rates rise by 10 basis points (

If rates rise by 10 basis points (bp bp) ) ΔΔy = y = ( 1% ) ( 1% )

Change in value = D*·∆y = ( 10 ) ( 1% ) = ( 1% )

Futures price change =

Futures price change = ∆P ∆P = = ( $90 ) ( 1% ) = $0.9 ( $90 ) ( 1% ) = $0.9

(i.e., from $90 to $89.10)

The gain on each short contract = 1,000 * $0.90 = $900

Price value of a basis point (PVBP) =

$900 / 10

$900 / 10 bp bp = $90 = $90

Hedge Ratio: Text Example

H =

=

PVBP for the portfolio PVBP for the hedge vehicle

$9,000

100

=

$90 per contract = 100 contracts

100 T-Bond futures contract will serve to offset the portfolio’s exposure to interest rate fluctuations The hedged position (long portfolio + short futures) has a PVBP of zero.

Practice Problems

BKM Ch 14: 3, 4, 5, 8a, 9, 10, 14, 22

BKM Ch 15:

Concept check: 8

Concept check: 8 9; 9;

End of chapter: 6, 14, CFA: 4, 10.

BKM Ch 16:

Concept check: 1

Concept check: 1 2; 2;

End of chapter: 2

End of chapter: 2 6, CFA: 3a 6, CFA: 3a 3c 3c.