l12: fixed income securities1 lecture 12: fixed income securities the following topics will be...

L12: Fixed Income Securities 1

Lecture 12: Fixed Income Securities

• The following topics will be covered:• Discount Bonds

• Coupon Bonds

• Interpreting the Term Structure of Interest Rates

• Basic of Term Structure Models

Materials from Chapter 10 and 11 (briefly) of CLM


Zero-coupon Bonds – basic notations

• For zero-coupon bonds, the yield to maturity is the discount rate which equates the present value of the bond’s payments to its price.

where Pnt is the time t price of a discount bond that makes a single payment of $1 at time t+n, and Ynt is the bond yield to maturity. We have,

• Expressed in log form, we have:

nnt

nt YP

)1(

1

)1

()1( nntnt PY

ntnt pn

y1


Yield Curve of Zero-coupon Bonds

• Term structure of interest rates is the set of yields to maturity, at a given time, on bonds of different maturities. Yield spread Snt=Ynt-Y1t, or in log term snt=ynt-y1t, measures the shape of the term structure.

• Yield curve plots Ynt or ynt against some particular date t.


Return for Discount Bonds (1)

• Define Rn,t+1 as the 1-period holding-period return on an n-period bond purchased at time t and sold at time t+1

• Writing in the log form, we have

• Holding period return is determined by the beginning-o-period yield (positively) and the change in the yield over the holding period (negatively).

11,1

1,11, )1(

)1()1(

ntn

nnt

nt

tntn Y

Y

P

PR

))(1(

)1(

1,1

1,11,11,

nttnnt

tnntnttntn

yyny

ynnyppr


Return for Discount Bonds (2)

• The log bond price today is the log price tomorrow minus the return today.

• We can solve this difference equation forward and get:

• We can also get:

The log yield to maturity on a zero-coupon bond equals the average log return period if the bond is held to maturity

1

01,

n

iitinnt rp

1,11, tntnnt prp

1

01,)/1(

n

iitinnt rny


Forward Rate• The forward rate is defined to be the return on the time t+n investment

of Pn+1,t/Pnt

where, in the forward rate, n refers to the number of periods ahead that the 1-period investment is to be made, and t refers to the date at which the forward rate is set.

nnt

ntn

nttnnt Y

Y

PPF

)1(

)1(

)/(

1)1(

1,1

,1

))(1(

)(

)1(

,1

,1,1

,1

,1

nttnnt

nttntn

nttn

tnntnt

yyny

yyny

nyyn

ppf


Coupon Bonds

• Coupon bonds can be viewed as a package of discount bonds

• There is no analytical solution for yield to maturity of coupon bonds

• Unlike the yield to maturity on a discount bond, the yield to maturity on a coupon bond does not necessarily equal the per-period return if the bond is held to maturity.– The yield to maturity equals the per-period return on the coupon bond held to

maturity only if coupons are reinvested at a rate equal to the yield to maturity.

• Two cases– Selling at par

– perpetuity

ncntcntcnt

cnt Y

C

Y

C

Y

CP

)1(

1...

)1()1( 2


Duration• Macaulay duration:

• See the example on page 402• Duration is the negative of the elasticity of a coupon bond’s

price with respect to its gross yield (1+Ycnt)

• Modified duration:

cnt

ncnt

n

i icnt

cnt P

Yn

Yi

C

D)1()1(1

cnt

cnt

cnt

cntcnt P

Y

Yd

dPD

)1(

)1(

cntcnt

cnt

cnt

cnt

PdY

dP

Y

D 1

)1(


Immunization• Implications: firms with long-term zero-coupon liabilities,

such as pension obligations, they may wish to match or immunize these liabilities with coupon-bearing Treasury bonds.– Zero-coupon Treasury bonds are available, they may be unattractive

because of tax clientele and liquidity effects, so the immunization remain relevant.

• If there is a parallel shift in the yield curve so that bond yields of all maturities move by the same amount, then a change in the zero-coupon yield is accompanied by an equal change in the coupon bond yield


Limitations

• A parallel shift of the term structure

• Works for small change in interest rates

• Cash flows are fixed and don’t change when interest rate changes.

– Callable securities

cntcnt

cnt

PdY

PConvexity

12

2

2)(*2

1)_mod( cncn

cnt

cnt dYconvexitydYDurationifiedP

dP


Loglinear Model for Coupon Bonds

• Starting from the loglinear approximate return formula, we have

cnttnctnc pcpkr )1(1,1,1,,

1,1,1,, )1( tnccncntcntnc yDyDr


Estimating Zero-coupon Term Structure

• If the prices of discount bonds P1…Pn maturing at each coupon date is known, then the price of a coupon bond is:

• If coupon bond prices are known, then we can get the implied zero-coupon term structure:

)1(...21 CPCPCPP ncn

C

CPCPPP

C

PP

ncnn

c

1

...

...1

11

11


Spline Estimation• When there are more than one price for each maturity,

statistical methods should be used. One way is regression:

• In practice the term structure of coupon bonds is usually incomplete. McCulloch (1971, 1975) suggest to write Pn as a function of maturity P(n):

• Assume P(n) to be a spline function. The term "spline" is used to refer to a wide class of functions that are used in applications requiring data interpolation and/or smoothing.

iiniinc uCPCPCPPii

)1(...21

J

j jjn nfanPP1

)(1)(


Tax Effect

• US Treasury bond coupons are taxed as ordinary income while price appreciation on a coupon bearing bond purchased at a discount is taxed as capital

• Thus there is a tax effect

• Page 411, CLM


Pure Expectation Hypothesis (PEH)• PEH

)]1)...(1)(1[()1( 111,11 nttttn

nt YYYEY

]1[)1(

)1(1 1,11

,1,1

nttntn

nnt

tn YEY

YF


Alternatives to Pure Expectation Hypothesis

• Expectation hypothesis– Considering term premia

• Preferred habitat– Different lenders and borrowers may have

different preferred habitats

• Time varying of term premia


Term Structure Models -- Motivations

• Starting from the general asset pricing condition introduces:

1=Et[(1+Ri,t+1)Mt+1]

• Fixed-income securities are particularly easy to price. When a fixed-income security has deterministic cash flows, it covaries with the stochastic discount factor only because there is time-variation in discount factors.

Pnt=Et[Pn-1,t+1Mt+1]

• It can be solved forward to express the n-period bond price as

Pnt=Et[Pn-1,t+1Mt+1]


Affine-Yield Models

• Assume that the distribution of the stochastic discount factor Mt+1 is conditionally lognormal

• Take logs of Pnt=Et[Pn-1,t+1Mt+1], we have

l12: fixed income securities1 lecture 12: fixed income securities the following topics will be...

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