Additional Praise for Fixed Income Securities Tools for Today’s Markets, 2nd Edition phần 5 docx

Figure 10.2 graphs three term structures of spot rates: one with no volatility around the expectation of 10%, one with a volatility of 200basis points a year the tree of the first example

structure as given are called arbitrage-free models of the term structure.

Another approach, to be described in this and subsequent chapters, is toderive the risk-neutral process from assumptions about the true interestrate process and about the risk premium demanded by the market for bear-ing interest rate risk Models that follow this approach do not necessarily

match the initial term structure and are called equilibrium models The

benefits and weaknesses of each class of models are discussed throughoutChapters 11 to 13

This chapter describes how assumptions about the true interest rateprocess and about the risk premium determine the level and shape of theterm structure For equilibrium models an understanding of the relation-ships between the model assumptions and the shape of the term structure isimportant in order to make reasonable assumptions in the first place Forarbitrage-free models an understanding of these relationships reveals theassumptions implied by the market through the observed term structure.Many economists might find this chapter remarkably narrow Aneconomist asked about the shape of the term structure would undoubtedlymake reference to macroeconomic factors such as the marginal productiv-ity of capital, the propensity to save, and expected inflation The moremodest goal of this chapter is to connect the dynamics of the short-termrate of interest and the risk premium with the shape of the term structure.While this goal does fall short of answers that an economist might provide,

it is more ambitious than the derivation of arbitrage restrictions on bondand derivative prices given the prices of a set of underlying bonds.

The first sections of this chapter present simple examples to illustratethe roles of interest rate expectations, volatility and convexity, and riskpremium in the determination of the term structure A more general, math-ematical description of these effects follows Finally, an application illus-trates the concepts and describes the magnitudes of the various effects inthe context of the U.S Treasury market

EXPECTATIONS

The word expectations implies uncertainty Investors might expect the year rate to be 10%, but know there is a good chance it will turn out to be8% or 12% For the purposes of this section alone the text assumes awayuncertainty so that the statement that investors expect or forecast a rate of10% means that investors assume that the rate will be 10% The sectionsfollowing this one reintroduce uncertainty

one-To highlight the role of interest rate forecasts in determining the shape

of the term structure, consider the following simple example The one-yearinterest rate is currently 10%, and all investors forecast that the one-yearinterest rate next year and the year after will also be 10% In that case, in-vestors will discount cash flows using forward rates of 10% In particular,the price of one-, two-, and three-year zero coupon bonds per dollar facevalue (using annual compounding) will be

Now assume that the one-year rate is still 10%, but that all investorsforecast the one-year rate next year to be 12% and the one-year rate in twoyears to be 14% In that case, the one-year spot rate is still 10% The two-

year spot rate, rˆ(2), is such that

(10.4)

Solving, rˆ(2)=10.995% Similarly, the three-year spot rate, rˆ(3), is such that

(10.5)

Solving, rˆ(3)=11.998% Hence, the evolution of the one-year rate from 10%

to 12% to 14% generates an upward-sloping term structure of spot rates:10%, 10.995%, and 11.988% In this case investors require rates above 10%when locking up their money for two or three years because they assume one-year rates will be higher than 10% No investor, for example, would buy atwo-year zero at a yield of 10% when it is possible to buy a one-year zero at10% and, when it matures, buy another one-year zero at 12%

Finally, assume that the one-year rate is 10%, but that investors cast it to fall to 8% in one year and to 6% in two years In that case, it iseasy to show that the term structure of spot rates will be downward-slop-

fore-ing In particular, rˆ(1)=10%, rˆ(2)=8.995%, and rˆ(3)=7.988%.

These simple examples reveal that expectations can cause the termstructure to take on any of a myriad of shapes Over short horizons, onecan imagine that the financial community would have specific views aboutthe future of the short-term rate The term structure in the U.S Treasurymarket on February 15, 2001, analyzed later in this chapter, implies thatthe short-term rate would fall for about two years and then rise again.1Atthe time this was known as the “V-shaped” recovery At first, the economywould continue to weaken and the Federal Reserve would continue to re-duce the federal funds target rate in an attempt to spur growth Then the

economy would rebound sharply and the Federal Reserve would be forced

to increase the target rate to keep inflation in check.2

Over long horizons the path of expectations cannot be as specific asthose mentioned in the previous paragraph For example, it would be diffi-cult to defend the position that the one-year rate 29 years from now will besubstantially different from the one-year rate 30 years from now On theother hand, one might make an argument that the long-run expectation ofthe short-term rate is, for example, 5% (2.50% due to the long-run real rate

of interest and 2.50% due to long-run inflation) Hence, forecasts can bevery useful in describing the level and shape of the term structure over shorttime horizons and the level of rates over very long horizons This conclusionhas important implications for extracting expectations from observed inter-est rates (see the application at the end of this chapter), for curve fitting tech-niques not based on term structure models (see Chapter 4), and for the use ofarbitrage-free models of the term structure (see Chapters 11 to 13)

VOLATILITY AND CONVEXITY

This section drops the assumption that investors believe their forecasts arerealized and assumes instead that investors understand the volatilityaround their expectations To isolate the implications of volatility on theshape of the term structure, this section assumes that investors are riskneutral so that they price securities by expected discounted value The nextsection drops this assumption

Assume that the following tree gives the true process for the one-yearrate:

1412

“U-Note that the expected interest rate on date 1 is 5×8%+.5×12% or 10%and that the expected rate on date 2 is 25×14%+.5×10%+.25×6% or10% In the previous section, with no volatility around expectations, flatexpectations of 10% imply a flat term structure of spot rates That is notthe case in the presence of volatility.

The price of a one-year zero is, by definition,1/1.10or 909091, ing a one-year spot rate of 10% Under the assumption of risk neutrality,the price of a two-year zero may be calculated by discounting the terminalcash flow using the preceding interest rate tree:

imply-Hence, the two-year spot rate is such that 82672=1/(1+rˆ(2))2, implying

that rˆ (2)=9.982%.

Even though the one-year rate is 10% and the expected one-year rate

in one year is 10%, the two-year spot rate is 9.982% The 1.8-basis pointdifference between the spot rate that would obtain in the absence of uncer-tainty, 10%, and the spot rate in the presence of volatility, 9.982%, is theeffect of convexity on that spot rate This convexity effect arises from the

mathematical fact, a special case of Jensen’s Inequality, that

(10.6)

Figure 10.1 graphically illustrates this equation The figure assumes that

there are two possible values for r, r Low and r High The curve gives values of

1/(1+r) for the various values of r The midpoint of the straight line ing 1/(1+r Low ) to 1/(1+r High) equals the average of those two values Under theassumption that the two rates occur with equal probability, this average

connect-equals the point labeled E[1/(1+r)] in the figure Under the same assumption, the point on the abscissa labeled E[1+r] equals the expected value of 1+r and the corresponding point on the curve equals 1/E[1+r] Clearly, E[1/(1+r)] is

E

11

11

11+

greater than 1/E(1+r) To summarize, equation (10.6) is true because the pricing function of a zero, 1/(1+r), is convex rather than concave.

Returning to the example of this section, equation (10.6) may be used

to show why the one-year spot rate is less than 10% The spot rate oneyear from now may be 12% or 8% According to (10.6),

The tree presented at the start of this section may also be used to price

a three-year zero The resulting price tree is

The three-year spot rate, such that 752309=1/(1+rˆ(3))3, is 9.952% fore, the value of convexity in this spot rate is 10%–9.952% or 4.8 basispoints, whereas the value of convexity in the two-year spot rate was only1.8 basis points

There-It is generally true that, all else equal, the value of convexity increaseswith maturity This will be proved shortly For now, suffice it to say that

the convexity of the price of a zero maturing in N years, 1/(1+r) N,

in-creases with N In other words, if Figure 10.1 were redrawn for the tion 1/(1+r)3, for example, instead of 1/(1+r), the resulting curve would be

func-more convex

Chapters 5 and 6 show that bonds with greater convexity perform ter when yields change a lot but mentioned that this greater convexity ispaid for at times that yields do not change very much The discussion inthis section shows that convexity does, in fact, lower bond yields Themathematical development in a later section ties these observations to-gether by showing exactly how the advantages of convexity are offset bylower yields

bet-The previous section assumes no interest rate volatility and, quently, yields are completely determined by forecasts In this section, withthe introduction of volatility, yield is reduced by the value of convexity So

conse-it may be said that the value of convexconse-ity arises from volatilconse-ity more, the value of convexity increases with volatility In the tree intro-duced at the start of the section, the standard deviation of rates is 200 basis

Further-1877193

points a year.3Now consider a tree with a standard deviation of 400 basispoints a year:

The expected one-year rate in one year and in two years is still 10% Spotrates and convexity values for this case may be derived along the same lines

as before Figure 10.2 graphs three term structures of spot rates: one with

no volatility around the expectation of 10%, one with a volatility of 200basis points a year (the tree of the first example), and one with a volatility

of 400 basis points per year (the tree preceding this paragraph) Note that

1814

3 Chapter 11 describes the computation of the standard deviation of rates implied

by an interest rate tree.

the value of convexity, measured by the distance between the rates ing no volatility and the rates assuming volatility, increases with volatility.Figure 10.2 also shows that the value of convexity increases with maturity.For very short terms and realistic volatility, the value of convexity isquite small Simple examples, however, must use short terms, so convexityeffects would hardly be discernible without raising volatility to unrealisticlevels Therefore, this section is forced to choose unrealistically largevolatility values The application at the end of this chapter uses realisticvolatility to present typical convexity values.

Risk-averse investors demand a return higher than 10% for the year zero over the next year This return can be effected by pricing thezero coupon bond one year from now at less than the prices of 1/1.14or.877193 and 1/1.06 or 943396 Equivalently, future cash flows could bediscounted at rates higher than the possible rates of 14% and 6% Thenext section shows that adding, for example, 20 basis points to each ofthese rates is equivalent to assuming that investors demand an extra 20basis points for each year of modified duration risk Assuming this is in-

two-deed the fair market risk premium, the price of the two-year zero would

be computed as follows:

(10.11)

First, this is below the price of 827541 obtained in equation (10.9) by suming that investors are risk-neutral Second, the increase in the discount-ing rates has increased the expected return of the two-year zero In oneyear, if the interest rate is 14% then the price of a one-year zero will be

as-1/1.14or 877193 If the interest is 6%, then the price of a one-year zero will

be 1/1.06or 943396 Therefore, the expected return of the two-year zeropriced at 826035 is

(10.12)

Hence, recalling that the one-year zero has a certain return of 10%, therisk-averse investors in this example demand 20 basis points in expectedreturn to compensate them for the one year of modified duration risk in-herent in the two-year zero.5

Continuing with the assumption that investors require 20 basis pointsfor each year of modified duration risk, the three-year zero, with its ap-proximately two years of modified duration risk,6 needs to offer an ex-pected return of 40 basis points The next section shows that this return

5 The reader should keep in mind that a two-year zero has one year of interest rate risk only in this stylized example: It has been assumed that rates can move only once a year In reality rates can move at any time, so a two-year zero has two years

of interest rate risk.

6 See the previous footnote.

can be effected by pricing the three-year zero as if rates next year are 20basis points above their true values and as if rates the year after next are 40basis points above their true values To summarize, consider the followingtwo trees If the tree to the left depicts the actual or true interest rateprocess, then pricing with the tree to the right provides investors with arisk premium of 20 basis points for each year of modified duration risk Ifthis risk premium is, in fact, embedded in market prices, then by definition,the tree to the right is the risk-neutral interest rate process.

True Process Risk-Neutral Process

The text now verifies that pricing the three-year zero with the tral process does offer an expected return of 10.4%, assuming that ratesactually move according to the true process

risk-neu-The price of the three-year zero can be computed by discounting usingthe risk-neutral tree:

To find the expected return of the three-year zero over the next year,proceed as follows Two years from now the three-year zero will be a one-

1844595

year zero with no interest rate risk.7Therefore, its price will be determined

by discounting at the actual interest rate at that time: 1/1.18or 847458, 1/1.10

or 909091, and 1/1.02or 980392 One year from now, however, the year zero will be a two-year zero with one year of modified duration risk.Therefore, its price at that time will be determined by using the risk-neutralrates of 14.20% and 6.20% In particular, the two possible prices of thethree-year zero in one year are

Continuing with the assumption of 400 basis point volatility, Figure10.3 graphs the term structure of spot rates for three cases: no risk pre-mium, a risk premium of 20 basis points per year of modified durationrisk, and a risk premium of 40 basis points In the case of no risk premium,the term structure of spot rates is downward-sloping due to convexity Arisk premium of 20 basis points pushes up spot rates of longer maturitywhile convexity pulls them down In the short end the risk premium effectdominates and the term structure is mildly upward-sloping In the long endthe convexity effect dominates and the term structure is mildly downward-sloping The next section clarifies why risk premium tends to dominate in

.769067= .5 847458( +.909091 1 142)

7 This is an artifact of this example in which rates change only once a year.

the short end while convexity tends to dominate in the long end Finally, arisk premium as large as 40 basis points dominates the convexity effect,and the term structure of spot rates is upward-sloping The convexity effect

is still evident, however, from the fact that the curve increases more rapidlyfrom one to two years than from two to three years

Just as the section on volatility uses unrealistically high levels ofvolatility to illustrate its effects, this section uses unrealistically high levels

of the risk premium to illustrate its effects The application at the end ofthis section focuses on reasonable magnitudes for the various effects in thecontext of the U.S Treasury market

Before closing this section, a few remarks on the sources of an interestrate risk premium are in order Asset pricing theory (e.g., the Capital AssetPricing Model, or CAPM) teaches that assets whose returns are positivelycorrelated with aggregate wealth or consumption will earn a risk premium.Consider, for example, a traded stock index That asset will almost cer-tainly do well if the economy is doing well and poorly if the economy is do-ing poorly But investors, as a group, already have a lot of exposure to theeconomy To entice them to hold a little more of the economy in the form

of a traded stock index requires the payment of a risk premium; that is, theindex must offer an expected return greater than the risk-free rate of re-turn On the other hand, say that there exists an asset that is negatively

correlated with the economy Holdings in that asset allow investors to duce their exposure to the economy As a result, investors would accept anexpected return on that asset below the risk-free rate of return That asset,

re-in other words, would have a negative risk premium

This section assumes that bonds with interest rate risk earn a riskpremium In terms of asset pricing theory, this is equivalent to assumingthat bond returns are positively correlated with the economy or, equiva-lently, that falling interest rates are associated with good times One ar-gument supporting this assumption is that interest rates fall wheninflation and expected inflation fall and that low inflation is correlatedwith good times

The concept of a risk premium in fixed income markets has probablygained favor more for its empirical usefulness than for its theoretical solid-ity On average, over the past 70 years, the term structure of interest rateshas sloped upward.8While the market may from time to time expect thatinterest rates will rise, it is hard to believe that the market expects interestrates to rise on average Therefore, expectations cannot explain a termstructure of interest rates that, on average, slopes upward Convexity, ofcourse, leads to a downward-sloping term structure Hence, of the three ef-fects described in this chapter, only a positive risk premium can explain aterm structure that, on average, slopes upward

An uncomfortable fact, however, is that over earlier time periods theterm structure has, on average, been flat.9Whether this means that an in-terest rate risk premium is a relatively recent phenomenon that is here tostay or that the experience of persistently upward-sloping curves is onlypartially due to a risk premium is a question beyond the scope of thisbook In short, the theoretical and empirical questions with respect to theexistence of an interest rate risk premium have not been settled

A MATHEMATICAL DESCRIPTION OF EXPECTATIONS,

CONVEXITY, AND RISK PREMIUM

This section presents an approach to understanding the components of turn in fixed income markets While the treatment is mathematical, the aim

re-is intuition rather than mathematical rigor

8 See, for example, Homer and Sylla (1996), pp 394–409.

9 See, for example, Homer and Sylla (1996), pp 394–409.

Let P(y, t; c) be the price of a bond at time t with a yield y and a tinuously paid coupon rate of c A continuously paid coupon means that over a small time interval, dt, the bond makes a coupon payment of cdt.

con-No real bonds pay continuous coupons, but the assumption will make themathematical development of this section simpler without any loss of intu-ition Note, by the way, that the coupon rate is written after the semicolon

to indicate that the coupon rate is fixed

By Ito’s Lemma (a discussion of which is beyond the mathematical

scope of this book),

(10.16)

where dP, dy, and dt are the changes in price, yield, and time, respectively

over the next instant and σ is the volatility of yield measured in basis pointper year The two first-order partial derivatives, ∂P/∂y and ∂P/∂t, denote thechange in the bond price for a unit change in yield (with time unchanged)and the change in the bond price for a unit change in time (with yield un-changed), respectively, over the next instant Finally, the second-order par-tial derivative, ∂2P/ ∂y2, gives the change in ∂P/∂yfor a unit change in yield(with time unchanged) over the next instant Dividing both sides of (10.16)

by price,

(10.17)

Thus, equation (10.17) breaks down the return from bond price changes overthe next instant, dP/P, into three components This equation can be written in amore intuitive form by invoking several facts from throughout this book.First, recall from Chapter 3 that, with an unchanged yield, the totalreturn of a bond over a coupon interval equals its yield multiplied by thetime interval Appendix 10A proves the continuous time equivalent ofthat statement:

(10.18)

In words, the yield equals the return of the bond in the form of price preciation plus the return in the form of coupon income Rearranging(10.18) slightly,

ap-y P

P t

c P

2 2 2

σ

A Mathematical Description of Expectations, Convexity, and Risk Premium 207

Second, Chapter 6 shows that modified duration and yield-based

con-vexity, written here as D and C, respectively, may be written as

(10.20)

(10.21)Substituting equations (10.19), (10.20), and (10.21) into (10.17),

(10.22)

The left-hand side of this equation is the total return of the bond—that

is, its capital gain, dP, plus its coupon payment, cdt, divided by the initial

price The right-hand side of (10.22) gives the three components of totalreturn The first component equals the return due to the passage of time—that is, the return to the bondholder over some short time horizon if yieldsremain unchanged The second and third components equal the return due

to change in yield The second term says that increases in yield reduce bondreturn and that the greater the duration of the bond, the greater this effect.This term is perfectly consistent with the discussion of interest rate sensitiv-ity in Part Two of the book

The third term on the right-hand side of equation (10.22) is consistentwith the related discussions in Chapters 5 and 6 Equation (5.20) showedthat bond return increases with convexity multiplied by the change in yield

squared Here, in equation (10.22), C is multiplied by the volatility of yield

instead of the yield squared By the definition of volatility and variance, ofcourse, these quantities are very closely related: Variance equals the ex-pected value of the yield squared minus the square of the expected yield.Equation (5.20) implied that positive convexity increases returnwhether rates rise or fall Equation (10.22) implies the same thing Also,Chapters 5 and 6 concluded that the greater the change in yield, the greaterthe performance of bonds with high convexity relative to bonds with lowconvexity Similarly, equation (10.22) shows that the greater the volatility

P y

P y

= − ∂

∂1

∂

of yield, the greater this convexity-induced advantage The text soon cusses the cost of this increased return.

dis-To draw conclusions about the expected returns of bonds with ent duration and convexity characteristics, it will prove useful to take theexpectation of each side of (10.22), obtaining

differ-(10.23)

Equation (10.23) divides expected return into its mathematical nents These components are analogous to those in equation (10.22): a re-turn due to the passage of time, a return due to expected changes in yield,and a return due to volatility and convexity To develop equation (10.23)further, the analysis must incorporate the economics of expected return.Risk-neutral investors demand that each bond offer an expected returnequal to the short-term rate of interest The interest rate risk of one bondrelative to another would not affect the required expected returns Mathe-matically,

compo-(10.24)

Risk-averse investors demand higher expected returns for bonds with moreinterest rate risk Appendix 10A shows that the interest rate risk of a bondover the next instant may be measured by its duration and that risk-averseinvestors demand a risk premium proportional to duration In the context

of this section, where yield is the interest rate factor, risk may be measured

by modified duration Letting the risk premium parameter be λ, the pected return equation becomes

Another useful way to think of the risk premium is in terms of the Sharpe

ratio of a security, defined as its expected excess return (i.e., its expected

E dP P

return above the short-term interest rate), divided by the standard deviation

of the return Since the random part of a bond’s return comes from its tion times the change in yield, the standard deviation of the return equals theduration times the standard deviation of the yield Therefore, the Sharpe ratio

dura-of a bond, S, may be written as

(10.26)

Comparing equations (10.26) and (10.25), one can see that S=λ/σ So, tinuing with the numerical example, if the risk premium is 10 basis pointsper year and if the standard deviation of yield is assumed to be 100 basispoints per year, then the Sharpe ratio of a bond investment is 10/100or 10%.Equipped with an economic description of expected returns, the textcan now draw conclusions about the determination of yield Substitute theexpected return equation (10.25) into the breakdown of expected returngiven by equation (10.23) to see that

con-(10.27)

Equation (10.27) mathematically describes the determinants of yieldpresented in this section The effect of expectations is given by the terms of

r and E[dy] For intuition, let y' denote the yield of the bond one instant

from now, let ∆y⬅y'–y, and let ∆t denote the time interval Then, the

ex-pectations terms of (10.27) alone say that

bond Finally, the greater a bond’s duration, the more its yield is determined

by expected future rates relative to the current short-term rate

The risk premium term of equation (10.27) shows an effect on yield of

Dλ As illustrated by the examples of this chapter, yield increases with thesize of the required risk premium and with the interest rate risk (i.e., dura-tion) of the bond

The discussion of the risk premium in the previous section and theconstruction of the risk-neutral trees in Chapter 9 show that pricing bonds

as if the short-term rate drifted up by a certain amount each year has thesame effect as a risk premium Inspection of equation (10.27) more for-mally reveals this equivalence As mentioned in the context of equation(10.29), the expected change in yield is driven by expected changes in theshort-term rate Increasing the expected yield by 10 basis points per yearimplies increasing the expected short-term rate by 10 basis points per year.Hence, equation (10.27) says that increasing the risk premium, λ, by afixed number of basis points is empirically indistinguishable from increas-ing the expected short-term rate by the same number of basis points peryear From a data perspective this means that the term structure at anygiven time cannot be used to distinguish between market expectations ofrate changes and risk premium From a modeling perspective this meansthat only the risk-neutral process is relevant for pricing Dividing the driftinto expectations and risk premium might be very useful in determiningwhether the model seems reasonable from an economic point of view, butthis division has no pricing implications

The term –(1/2)Cσ2in equation (10.27) gives the effect of convexity onyield Recalling from Chapter 6 that a bond’s convexity increases with ma-turity, this term shows that the convexity effect on yield increases with ma-turity and with interest rate volatility, as illustrated in the simple examplesgiven earlier

Recall from Part Two that duration increases more or less linearly withmaturity while convexity increases more or less with maturity squared.This observation, combined with equation (10.27), implies that, holdingeverything else equal, as maturity increases, the convexity effect eventuallydominates the risk premium effect However, as will be discussed in thenext section and in next few chapters, the volatility of yields tends to de-cline with maturity The 10-year yield, for example, is more volatile thanthe 30-year yield Therefore, as maturity increases, the increase in the con-vexity effect in (10.27) may be muted by falling volatility

A Mathematical Description of Expectations, Convexity, and Risk Premium 211

Equation (10.23) shows that the expected return of a bond is enhanced

by its convexity in the quantity (1/2)Cσ2 per unit time But the convexityterm in equation (10.27) shows that the yield and, therefore, the return due

to the passage of time are reduced by exactly that amount Hence, asclaimed in Chapters 5 and 6 and as mentioned earlier in this chapter, a bondpriced by arbitrage offers no advantage in expected return due to its con-vexity In fact, the expected return condition (10.25) ensured that this had

to be so None of this means, of course, that the return profile of bondswith different convexity measures will be the same Bonds with higher con-vexity will perform better when yields change a lot, while bonds with lowerconvexity will perform better when yields do not change by much

APPLICATION: Expectations, Convexity, and Risk Premium in the U.S Treasury Market

on February 15, 2001

Figure 10.4 shows four curves The uppermost curve is the par yield curve on February 15,

2001 These par yields are computed from the spot rates constructed in Chapter 4 The other three curves break down the par yields into the components discussed in this section: expectations, risk premium, and convexity.

As shown in the previous section, convexity impacts the yield of a bond by –( 1 / 2)Cσ2 The convexity of a particular par bond may be computed using the formulas given in Chapter 6.

FIGURE 10.4 Expectations, Convexity, and Risk Premium Estimates in the Treasury Market, February 15, 2001

Risk Premium

Convexity

Choosing the level of volatility to input into the convexity effect, however, requires some comment The desired quantity, the volatility of the yield in question, is unknown The most common substitute choices are one’s best guess, recent historical volatility, or implied volatility The relative merits of these choices will be discussed in Chapter 12, but Figure 10.4 uses implied volatilities from the relatively liquid short-term options on 2-, 5-, 10-, and 30- year Treasury securities Table 10.1 lists these implied volatilities as of February 15, 2001 Notice that the term structure of volatilities slopes upward and then downward For the purposes of Figure 10.4, implied volatilities on bonds of intermediate maturities were assumed to be linear in the given volatilities For maturities less than two years, the two- year volatility was used So, for example, the convexity effect on a 20-year security is computed as follows The convexity of a 20-year par bond at a yield of 5.67% is about 194.5 Interpolating 91.5 basis points per year for a 10-year yield and 68.6 basis points per year for a 30-year yield gives an approximation for 20-year volatility of about 80 basis points per year Therefore, the magnitude of the convexity effect is estimated at about 62 basis points:

(10.30)

Figure 10.4 illustrates that the magnitude of the convexity effect increases with maturity When increasing maturity, the increase in bond convexity offsets the decrease in volatility Adding the convexity effect to the par yields leaves expectations and risk premium Given the observational equivalence of these two effects (see the previous section and Chapter 11), there is no scientific way to separate them by observing a given term struc- ture of yields Therefore, for the purposes of drawing Figure 10.4, several strong as- sumptions were made 10 First, the long-run expectation of the short-term rate is about

1

2

× ×( ) % = %

APPLICATION: Expectations, Convexity, and Risk Premium 213

TABLE 10.1 Volatilities Implied from Short-Dated Bond Options

Term Basis Point Volatility

decom-5%, corresponding to a long-run real rate of 2.50% and a long-run inflation rate of 2.50% Second, the expectation curve is relatively linear at longer maturities As men- tioned in the discussion of expectations in this chapter, while the market might expect the short-term rate to move toward some long-term level, it is hard to defend any ex- pected fluctuations in the short-term rate 20 or 30 years in the future Third, the risk premium is a constant While the risk premium may, in theory, depend on calendar time and on the level of rates, very little theoretical work has been done to justify these rela- tively complex specifications Fourth, the Sharpe ratio of bonds is not too far from his- torical norms.

As it turns out, a risk premium of 9 basis points per year satisfies these objectives relatively well, although not perfectly well The resulting expectations curve exhibits a dip and then a gradual increase to a long-run level The dip is perfectly acceptable, cor- responding to beliefs about near-term economic activity and the Federal Reserve’s likely responses to that activity On the other hand, that the expectation curve rises above 5%,

to a maximum of about 5.23%, before falling back to 5% violates, at least to some tent, the second objective of the previous paragraph Lastly, using the volatilities given

ex-in Table 10.1, the magnitude of the risk premium gives Sharpe ratios rangex-ing from 9.4% for 5-year bonds to 13.1% for 30-year bonds These values are in the range of historical plausibility.

Decompositions of the sort described here are useful in forming opinions about which sectors of a bond market are rich or cheap Say, for example, that one accepts the decom- position presented here but does not accept that the expected short-term rate 20 years from now can be so far above the expected short rate 10 and 30 years from now In that case, one must conclude that 20-year yields are too high relative to 10-year and 30-year yields, or, equivalently, that the 20-year sector is cheap This conclusion suggests purchas- ing 20-year Treasury bonds rather than 10- and 30-year bonds or, more aggressively, buy- ing 20-year bonds and shorting 10- and 30-year bonds This, in fact, is the trade suggested

in the trading case study in Chapter 8 (The trade is rejected there because the 20-year tor did not seem cheap relative to recent history.)

sec-APPENDIX 10A

PROOFS OF EQUATIONS (10.19) AND (10.25)

Proof of Equation (10.19)

Under continuous compounding, the present value of the continuously

paid coupon payments from time t to the maturity of the bond at time T is

Adding the present value of the final principal payment to the value of thecoupon flow, the price of the bond is

(10.32)

Note that this is the continuous time equivalent of equation (3.4)

Taking the derivative of (10.32) with respect to t,

(10.33)But, rearranging (10.32) shows that

(10.34)Finally, combining equations (10.33) and (10.34),

(10.35)or

(10.36)

as was to be proved

Proof of Equation (10.25)

This proof follows that of Ingersoll (1987) and assumes some knowledge

of stochastic processes and their associated notation The notation will bedescribed in Chapter 11

Let x be some interest rate factor that follows the process

Let P be the full price of some security that depends on x and time Then,

P Q PQ

x Q

Rearranging (10.44),

(10.45)

Equation (10.45) says that the expected return of any security above

the short-term rate divided by its duration with respect to the factor x must

equal some function λ This function cannot depend on any characteristic

of the security because (10.45) is true for all securities The function maydepend on the factor and time, although this book, for simplicity, assumesthatλ is a constant Rewriting (10.45), for each security it must be true that

(10.46)

The derivation here assumes there are no coupon payments, while thediscussion in the text accounts for a coupon payment Also, the derivationhere uses an arbitrary interest rate factor, while the discussion in the text takesthe yield of a particular bond as the factor This is somewhat inconsistentsince this derivation requires every security to have the same factor while thetext implies a result simultaneously valid for every bond at its own yield