principles of corporate finance

Principles of Corporate Finance

Chapter 4. Valuing bonds

Ciclo Profissional 2o Semestre / 2009

Graduacao em Ciencias Economicas

V. Filipe Martins-da-Rocha (FGV) Principles of Corporate Finance August, 2009 1 / 42

Topics covered

1 Using the PV formula to value bonds2 How bond prices vary with interest rates3 The term structure and YTM4 Explaining the term structure5 Real and nominal rates of interest


Long-term loans

If they need cash for long term investments, firms generally issuebondsGovernment also issue bondsThe interest rate on government bonds are benchmarks for allinterest ratesFinancial managers had better understanding how the governmentrates are determined and what happen when they change


What is a bond?

The bond is characterized byI a maturity period τI a face value FI an interest rate i

The bond pays a coupon C at every period before τ where

C = F × i

At maturity, the bond pays the last coupon and the face value Fwhich is called the principal

Remark

We should not confuse the interest rate i and the rate of return of thebond


An example

In Germany, the government issues long term bonds in euroswhere interest (or coupons) are paid annuallySuppose that in July 2006 you decided to buy a bond with e100face value and 5% interest (called 5% bund) maturing in July 2012Each year until 2012 you are entitled to an interest payment of0.05× e100 = e5e5 is the bond’s coupon


An example

The present value of a bond corresponds to its price if the bond ismarketedIf the bond is not marketed, one should take as the discount ratethe rate of return offered by securities with similar riskFor instance, in July 2006 other medium-term Germangovernment bonds offered a rate of return of about 3.8%Discounting the bond’s cash at 3.8% we get

PV =e5

(1 + 0.038)+

e5(1 + 0.038)2

+ . . .+e5

(1 + 0.038)5+

e105(1 + 0.038)6

= e106.33


An example

We can use the annuity formula to value the coupon paymentsAnd add the present value of the final payment

PV(bond) = PV(coupon payments) + PV(final payment)

= e5 ×[

10.038

− 10.038(1 + 0.038)6

]+

e100(1 + 0.038)6

= e26.38 + e79.95= e106.33


Yield to maturity

Consider a bond that is marketed, i.e., there is a price VIn that case we have an investment with the following cash flows

C0 C1 C2 . . . Cτ−1 Cτ

-V iF iF . . . iF (1 + i)F

The “time invariant” rate of return y associated to this investmentis defined by the implicit equation

V =iF

(1 + y)+

iF

(1 + y)2+ . . .+

iF

(1 + y)τ−1+

(1 + i)F(1 + y)τ

The internal rate of return y of the “bond investment” is calledthe bond’s yield to maturity (YTM)


Bonds in United States

The U.S. Treasury periodically raises money by auctioning newissues of bonds

I Some of the bonds mature in 30 yearsI Others, known as notes, mature in 10 years or lessI There are also short-term loans that mature in less than a year:

Treasury bills

Treasury have a face value of $1,000We only have to specify the interest (defining the coupon) and thematurity

I A 4% note maturing in 2010 is called “4s of 2010”Interest on Treasury bonds are paid semiannully

I A 4s of 2010 provides a coupon payment of 4%/2 = 2% of face valueevery six months

Once issued, Treasury bonds are widely traded through a networkof dealers


Wall Street Journal, June 2006



Consider the 4s of 2009 bought in June 2006The asked price of 97:11 is the price we need to pay to buy thebond from a dealer

I The price is quoted in 32nds rather than decimalsI A price of 97:11 means that each bond costs 97 and 11/32, i.e.,

97.34375% of the face value or $973.4375

The bid price is the price investors receive if they sell the bond toa dealerThe spread between the asked and bid price makes the living ofthe dealer


Example

Consider the 4s of 2009 bought in June 2006The cash flows are as follows

The price is expressed in % of the face valueIf the price 97.35% then the semiannual compounded yield of the4s of 2009 in June 2006 is 4.96%

$973.5 =$20

(1 + 0.0248)+ . . .+

$20(1 + 0.0248)5

+$1020

(1 + 0.0248)6


How bond prices vary with interest rates?

Suppose that investors now demand a yield of 3% (instead of4.96%) on three-year Treasury bondsWhat would be the price of the 4s of 2009 in June 2006?

PV =$20

(1 + 0.015)+ . . .+

$20(1 + 0.015)5

+$1020

(1 + 0.015)6= $1, 028.49

or 102.85% of the face valueThe lower interest rate results in a higher bond priceBond investors cross their fingers that market interest rates willfall, so that the price of their securities will riseAny change in interest rates is likely to have only a modest effecton the value of near-term cash flowsThus the price of long-term bonds is affected more by changinginterest rates than is the price of short-term bonds


How bond prices vary with interest rates?


Duration: definition

What does it mean a “long-term” or “mid-term” bond?Every year there is a coupon paymentsBond analysts introduced the concept of duration

Definition

The duration of a bond is the “average” time to each payment asdefined by the formula

Duration = 1 × PV(C1)V

+ 2 × PV(C2)V

+ . . .+ τ × PV(Cτ )V

where V is the value of the bond, Ct is the payment in period t and τis the number of periods to maturity


Duration: example

Consider a 3-year 10% bond with face value $1,000Assume the yield to maturity (YTM) is 5%

Duration = (1 × 0.084) + (2 × 0.080) + (3 × 0.836) = 2.753 years

Take know another 3-year bond with the same maturity but thecoupon payment is 4%The duration of this bond is 2.884 yearsThe first two years’ coupon payments account for a smallerfraction of the value, in this sense this bond is a longer bond


Volatility

Definition

The volatility of a bond is its relative variation with respect to theyield, i.e.,

Volatility(y) = − 1V

∂V

∂y(y)

The 4% bond has the greater volatility and also the longerduration

Proposition

Volatility(y) =Duration(y)

1 + y


Volatility

The 30-year bond has a much longer duration than the 3-year bondCorrespondingly the 30-year bond is more volatile


Term structure of interest rates

The effect of interest rate changes is not the same on all types ofbonds

Between 1992 and 2000, short-term interest rates nearly doubledWhile long-term interest rates declined



Consider a simple loan that pays $1 at time 1

PV =1

1 + r1

where the discount rate r1 corresponds to today’s rate for aone-period loan: it is called today’s 1-period spot rate

If we have a loan that pays $1 at both time 1 and 2, its presentvalue is then

PV =1

1 + r1+

1(1 + r2)2

where r2 is today’s 2-period spot rate

The sequence (r1, r2, . . .) is one way of expressing the termstructure of interest ratesThe yield to maturity (YTM) of a bond may be regarded as an“average” of spot rates of interest and may hide useful information



Comparing the yields of two bonds is potentially misleadingIt is 2009 and we are contemplating an investment in U.S.Treasuries

Does the higher yield on the 5s of 2014 mean that they are abetter buy?



We can prove that both bonds are “fairly prices”In the sense that their price correspond to the present value for aspecific common term structure

The YTM is misleading since the same rate is used to discount allpayments of the bond, implicitly assuming that the term structureis flat


Stripped bonds

A bond making a single payment at time t is called a stripThe prices of strips are shown each day in the financial press (June2006 below)

Investors required a higher interest rate for lending for 10 yearsrather than 1


Forward interest rate

Suppose that the 1-year spot rate at time 0 is r1 = 5%Suppose that the 2-year spot rate at time 0 is r2 = 6%One dollar invested in a 1-year Treasury trip grows to $1.05If instead the dollar is invested in a 2-year strip, by the end of the2 years it would have grown to $1 × (1 + 0.06)2 = $1.1236By keeping your money invested for two years rather than one, itgrows from $1.05 to $1.1236, an increase of 7.01%This extra 7.01% is termed the forward interest rate f2 definedby

(1 + r2)2 = (1 + r1)(1 + f2)


The expectations theory

At time 1, one may invest the return $1.05 of the 1-year strip in a1-year strip paying in year 2Denote by 1r2 the expected (at the intial period year 0) future1-year strip available for trade in time 1 and paying in year 2The expectations theory claims that we are at equilibrium andinvestors perfectly anticipate future interest rateIn that case, by non-arbitrage we must have

f2 = 1r2


Forward interest rate and the expectations theory



According to this theory, the only reasons for an upward-sloping(declining) term structure is that investors expect short-terminterest rates to rise (fall)The expectations theory also implies that investing in a successionof short-term bonds gives exactly the same expected return asinvesting in long-term bondsThese days the expectations theory has few strict adherentsMost economists believe that expectations about future interestrates have an important effect of term structureBut it is not the whole truth



During the period 1900-2006, the return on long-term U.S.Treasury bonds was on average 1.2 percentage points higher thanthe return on short-term Treasury billsA possible explanation is that expectations where overestimatingfuture increases in interest ratesIt is more likely that investors wanted some extra return forholding long bonds and that on average they got itIf so the expectations theory is not sufficient to explain termstructureWhat are the other possible explanations? What determines theshape of the term structure? What does expectations theory leaveout?


Risk and term structure

If you are confident about future level of spot interest rates, youwill simply choose the strategy that offers the highest interest rateIf you are not sure of your forecasts, you may well opt for a lessrisky strategy even if it means giving up some (expected butuncertain) returnLong-duration bonds are more volatile than those of short-termbondsFor some investors this extra volatility may not be a concern:

I pension funds and life insurance companies with long-term bonds

However the volatility of long-term bonds does create extra riskfor investors who do not have such long-term obligationsThese investors will be prepared to hold long-term bonds only ifthey offer the compensation of a higher return


Inflation and term structure

The cash flows on U.S. Treasury are certainHowever, we cannot be sure what the money will buy at maturityIt depends on the rate of inflationUncertainty about inflation may make it more risky to invest inlong-term bondsBy investing in successive short-term bonds, we do not knowexactly the interest rate at which we will be able to reinvest themoney at the end of each year, but at least we know that it willincorporate the latest information about inflation in the comingyearHere is another reason that long-term bonds may offer a riskpremium


Inflation

Several indexes are used to track the general level of pricesThe best known in U.S. is the Consumer Price Index, CPI

I It measures the number of dollars that it takes to pay for a typicalfamily’s purchases

The change of CPI from one year to another measures the rate ofinflationIn U.S. the rate of inflation touched a peak at the end of WW1,when it reached 21%In Germany, the peak was in 1923 which was more than20,000,000,000% (5% a day)In recent years, Japan and Hong Kong have both faced a problemof deflation


Annual U.S. inflation rates from 1900 to 2006


Average rates of inflation from 1900 to 2006


Real vs. nominal interest rates

Most bonds promise a fixed nominal rate of interestThe real payoff of bonds depend on the inflation rate

real cash flowt =nominal cash flowt

(1 + inflation rate)t

In terms of interest rates, we have

1 + rreal =1 + rnominal

1 + rinflation


U.K. bonds’ yields

The real yield has been much more stable than the nominal yield


Equilibrium real interest rate

The real interest that investors require is determined by the demandfor capital and the supply of savings

The demand for capital comes from governments and firms thatwant to invest in new projectsThe supply of savings comes from individuals who are willing toconsume tomorrow rather than todayThe equilibrium interest rate is the rate that produces a balancebetween the demand and supply


Fisher’s theory (1930)

A change in the expected inflation rate will cause the sameproportionate change in the nominal interest rateIt has no effect on the required real interest rate


Fisher’s theory (1930)

Suppose that consumers are equally happy with 100 apples todayor 105 apples in a year’s timeIn this case the real “apple” interest rate is 5%If the price of the apple is constant (no inflation) at say $1 each,then consumers are equally happy to receive $100 today or $105 atthe end of the yearSuppose now that the apple price is expected to increase by 10%to $1.10In that case consumers require a nominal rate of interest 15.50%

(1 + 0.155) = (1 + 0.05)(1 + 0.10)

However the real interest rate is unchanged


T-bills vs. inflation from 1953 to 2006


principles of corporate finance

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