InvestmentsLecture 7: Bond Pricing and the Term Structure of Interest Rates

Instructor: Chyi-Mei ChenCourse website: http://www.fin.ntu.edu.tw/∼cchen/

1. Consider a discrete-time economy where financial markets open at datest = 0, 1, 2, · · · . At date t < T , a fixed-income security (C,F, T ) is a debtinstrument which pays C at each date t+1, t+2, · · · , T−1, and C+F atdate T , where C is the coupon payment, F is the face value, par value,or pricinpal value, C

Fis the coupon rate, and T the maturity date of the

bond.1 Throughout this note the fixed-income securities are assumedto be default-free; the pricing problem of fixed-income securities subjectto default risks is still a research subject in finance.

Note that the coupon bond becomes a zero-coupon bond (or pure dis-count bond) if C = 0, and in the latter case we always make thenormalization F = 1, and denote the zero-coupon bond maturing atdate T by B(t, T − t). We shall also let B(t, T − t) denote the date-tprice of the zero-coupon bond B(t, T − t).

2. If zero-coupon bonds of all maturities τ = 1, 2, · · · , T − t are traded atdate t, then any coupon bond (C,F, T ) can be priced by no arbitrage.In fact, to rule out arbitrage opportunities, the date-t price of a couponbond (C,F, T ) must be equal to

F ·B(t, T − t) + CT−t∑j=1

B(t, j).

Note that here the zero-coupon bonds play a role similar to that playedby Arrow-Debreu securities in Lecture 5; that is, the zero-coupon bondsB(t, 1), B(t, 2), · · · , B(t, T − t) can be regarded as basis vectors thatspan the entire set of coupon bonds with maturity dates less than orequal to T . More precisely, given T , if we represent a default-free asset

1Here we identify a fixed-income security with a coupon bond. The three major classesof fixed-income securities in the U.S. are treasury securities, corporate notes and bonds,and mortgage-backed securities.

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with maturity less than or equal to T by its per-unit (non-negative)payoff generated at dates t+1, t+2, · · · , T , then the set of such default-free assets can be represented by ℜT−t

+ . In this case, each coupon bondwith maturity date less than or equal to T can be regarded as a positive,linear combination of the (T − t) standard basis vectors in ℜT−t, whichare exactly the zero-coupon bonds B(t, 1), B(t, 2), · · ·B(t, T−t). Hencethe set of zero-coupon bonds plays the same role as the set of Arrow-Debreu securities does in Lecture 5.2

Example 1 Suppose that T = 5 and t = 0, and markets are completeup to year 5. In the following table, Pj denotes the date-0 price of thecoupon bond described in the j + 1 row.3

Maturity C F P1-yr 30 1000 978.502-yr 40 1000 974.003-yr 60 1000 1001.404-yr 80 1000 1068.405-yr 55 1000 961.55

Note that the above 5 coupon bonds can be represented as 5 elements ofℜ5, with their z5×1 vectors being

z1 =

10300000

, z2 =

401040000

, z3 =

6060106000

, z4 =

80808010800

, z5 =

555555551055

.

2Most bonds traded in the real world are coupon bonds. Zero-coupon bonds made theirdebut in the U.S. in the 1980’s, but the set of traded zero-coupon bonds is inadequate tospan the the set of traded coupon bonds. The following example shows how we can recoverthe prices of the non-traded zero-coupon bonds from the set of traded coupon bonds.

3We assume here that coupon payments are made annually. In the U.S. and Japan,coupon payments are typically paid semi-annually, but in Europe, they are often paidannually.

2

Correspondingly, we have

Pj = z′j

B(0, 1)B(0, 2)B(0, 3)B(0, 4)B(0, 5)

, j = 1, 2, · · · , 5.

Solving, we have

B(0, 1) = 0.95, B(0, 2) = 0.90, B(0, 3) = 0.84, B(0, 4) = 0.79, B(0, 5) = 0.73.

These 5 prices will allow us to price any fixed-income security withmaturity date less than or equal to T = 5.

3. If at date t one puts 1 dollar in a time deposit for τ = s − t periods(called the maturity of the deposit), then he will get (1+Yt(τ))

τ dollarsfor sure at date s, where note that Yt(τ) is the per-period (compound-ing) interest rate for this time deposit. (Note that he is not allowed towithdraw money from the deposit account earlier than date s.) Equiv-alently, if at date t he puts (1 + Yt(τ))

−τ dollars in a time deposit ofperiod τ , then he will get 1 dollar for sure at date s.4

The graph of the function Yt(·) is called the date-t yield curve, whichmaps τ (called the time-to-maturity) into the interest rate per periodYt(τ) required for a loan with maturity τ . Note that these interest ratesare spot rates, in the sense that they apply only to the loans that arecreated and executed at time t. In general, the date-(t+1) yield curveYt+1(·) will differ from the date-t yield curve Yt(·). This should not besurprising, because new information about the demand and supply ofloanable funds will arrive between date t and date t+ 1.

If zero-coupon bonds of all maturities are traded at date t, there is asecond way that one can get one dollar for sure at date s > t by tradingonly at date t, which is buying at date t 1 unit of the zero-coupon bond

4As a convention, from now on I shall put a˜on a variable or expression to emphasizethat it is random.

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maturing at date s. Recall that B(t, τ) is the date-t price of that bond,where τ = s− t. To rule out arbitrage opportunities, we must have

B(t, τ) = (1 + Yt(τ))−τ .

Because of this equation, the interest rate Yt(τ) is formally referredto as the date-t yield to maturity (or YTM) for the zero-coupon bondB(t, τ). Note that the YTM Yt(τ) is exactly the rate of return on thestrategy of buying the zero-coupon bond B(t, τ) at date t and holdingit till the maturity date t+ τ .

There is a third way of getting one dollar for sure at date s by makingmoves only at date t, which involves forward transactions. Considera lender and a borrower that sign a loan contract at date t promisingto lend and borrow respectively at date t′ ≥ t and to repay the loanat date s > t′. Such a contract is called a forward loan contract. Theinterest rate that appears in this forward contract, denoted by ft(t

′, s),is referred to as a forward rate, as opposed to the spot rates definedabove. By definition,

ft(t, t+ 1) = Yt(1).

Now, the third way of obtaining 1 dollar at date s by making movesonly at date t is as follows: one can (1) sign at date t a forward contractstating that he will lend (1 + ft(t

′, s))t′−s dollars at date t′ > t, and

be repaid at date s; and (2) buy at date t (1 + ft(t′, s))t

′−s units ofthe zero-coupon bond B(t, t′ − t). This strategy generates 1 dollar atdate s, because, by (2), a date-t′ cash inflow (1 + ft(t

′, s))t′−s will be

generated, which, by (1), must be lent out at the interest rate ft(t′, s)

for s− t′ periods, so that a final payoff of

(1 + ft(t′, s))t

′−s × (1 + ft(t′, s))s−t′ = 1

will be generated at date s. What is the date-t cost of this strategy?Signing the forward contract is costless, and so (1) costs nothing; butthere is a cost to implement (2), which is obviously

(1 + ft(t′, s))t

′−s ×B(t, t′ − t) = (1 + ft(t′, s))t

′−s × (1 + Yt(t′ − t))t−t′ .

This is the date-t cost of the third way to generate 1 dollar at date s!To rule out arbitrage opportunities, this cost must equal the cost of

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implementing the above second way of generating 1 dollar at date s.Hence we require

(1 + Yt(s− t))t−s = (1 + ft(t′, s))t

′−s × (1 + Yt(t′ − t))t−t′

= (1 + Yt((s− 1)− t))t−(s−1)(1 + ft(s− 1, s))−1

= (1 + Yt((s− 2)− t))t−(s−2)(1 + ft(s− 2, s− 1))−1(1 + ft(s− 1, s))−1

= · · · = [(1+ft(t, t+1))(1+ft(t+1, t+2)) · · · (1+ft(s−2, s−1))(1+ft(s−1, s))]−1.

That is, 1 + Yt(s − t) is the geometric mean of [1 + ft(t, t + 1)], [1 +ft(t+ 1, t+ 2)], · · · , [1 + ft(s− 1, s)].

Because each zero-coupon bond has a face value equal to 1 dollar, wealso refer to B(t, τ) as the τ -period discount factor at date t, and Yt(τ)as the corresponding discount rate.

4. The above discussions have the following implications. First, if giventhe current date t and a future date T , there are T − t linearly in-dependent coupon bonds traded at date t, then we can recover allB(t, τ); τ = 1, 2, · · · , T−t; and if we are given B(t, τ); τ = 1, 2, · · · , T−t, then we can price all coupon bonds with maturities less than or equalto date T . Second, knowing one of the following three is equivalent toknowing all three of them:

• The date-t yield curve Yt(·) defined for maturities less than orequal to T − t.

• The date-t forward rates up to the repaying date T .

• The date-t prices of all coupon bonds with maturity dates lessthan or equal to date T .

5. Now let us review the idea of YTM (yield to maturity) for a couponbond (C,F, T ). Let Pt be the date-t price of this coupon bond. Thenwe can first compute P0 using the formula

P0 = F ·B(0, T ) + CT∑

j=1

B(0, j).

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Once we obtain P0, we can define the YTM for this coupon bond asthe solution to the equation P0 = g(x), where x is the unknown, and

g(x) ≡ F · (1 + x)−T + CT∑

j=1

(1 + x)−j;

that is, g(x) is the date-0 present value of the stream of cash flowsgenerated by holding 1 unit of the coupon bond till date T , using x asthe discount rate. The discount rate x such that g(x) = P0 is referred toas the coupon bond’s date-0 YTM. Those who have taken a corporatefinance course will recognize that the YTM so defined is simply theinternal rate of return (IRR) generated by the strategy of buying thecoupon bond at date 0 and holding it till the maturity date T . Notethat in the special case where C = 0 and F = 1, the above YTM equalsexactly the Y0(T ) defined in section 3.

We claim that the function g(·) defined above is such that g′ < 0 < g′′;that is, g(·) is a decreasing and convex function. To see this, note that

g′(x) = −T · F · (1 + x)−T−1 − CT∑

j=1

j(1 + x)−j−1 < 0,

and

g′′(x) = T (T + 1)F · (1 + x)−T−2 + CT∑

j=1

j(j + 1)(1 + x)−j−2 > 0.

Now, we claim that

g(C

F) = F.

To see this, note that

g(C

F) = F(1+C

F)−T+

C

F

T∑j=1

(1+C

F)−j = F(1+C

F)−T+

CFCF

[1−(1+C

F)−T ] = F.

We hence reach the conclusion that

P0 = g(YTM) > (respectively, <)g(C

F) = F ⇔ YTM < (respectively, >)

C

F.

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We say that the coupon bond is sold at a premium (respectively, adiscount) at date 0 if P0 > F (respectively, P0 < F ). The above resultshows that the bond will be sold at a premium (respectively, a discount)at date 0 if and only if the coupon rate C

Fis greater (respectively, less)

than its date-0 YTM.

6. Now let us define the yield curve for coupon bonds. From the lastsection, we see that the YTM of a coupon bond depends on C,F , andT . We argue that in one particular situation, the YTM depends onlyon T , so that we have a yield curve for coupon bonds. This is thesituation when we restrict attention to the coupon bonds that are soldat face value.5 From the last section, we know that this happens whenthe date-0 YTM= C

F. In this case, letting x0(T ) denote the date-0

YTM for the coupon bond (C,F, T ), we have

P0 = F = F(1 + x0(T ))−T + x0(T )

T∑j=1

(1 + x0(T ))−j

= F(1 + Y0(T ))−T + x0(T )

T∑j=1

(1 + Y0(j))−j,

so that x0(T ) is the unique solution to the following equation (with xbeing the unknown):

1 = (1 + x)−T + xT∑

j=1

(1 + x)−j = (1 + Y0(T ))−T + x

T∑j=1

(1 + Y0(j))−j

⇒ x0(T ) =1− (1 + Y0(T ))

−T∑Tj=1(1 + Y0(j))−j

.

where we have taken the yield curve Y0(·) as known data. In this case,given Y0(·), we can obtain a par yield curve x0(·) for the date-0 couponbonds (with various times to maturity) that are sold at face value atdate 0.

The following example shows that the par yield curve x0(·) of thecoupon bonds will be upward sloping if the yield curve (of the zero-coupon bonds) Y0(·) is, and in that case x0(·) lies below Y0(·).

5In this case, the YTM is called the par-coupon yield of the coupon bonds.

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Let T = 3, t = 0, Y0(1) = 0.1, Y0(2) = 0.15, Y0(3) = 0.2. Letx0(1), x0(2), x0(3) denote the date-0 YTM for coupon bonds with T =3. Then we have

1 =1 + x0(1)

1 + Y0(1),

1 =1 + x0(2)

[1 + Y0(2)]2+

x0(2)

1 + Y0(1),

1 =1 + x0(3)

[1 + Y0(3)]3+

x0(3)

[1 + Y0(2)]2+

x0(3)

1 + Y0(1).

It can be easily verified that x0(3) > x0(2) > x0(1), but x0(t) ≤ Y0(t),for t = 1, 2, 3.

Now it is equally easy to verify that the yield curve of the coupon bondswill be downward sloping if the yield curve (of the zero-coupon bonds)is, and in that case the former lies above the latter.

Note also that when Y0(·) is flat, then x0(·) is also flat, and in fact thesetwo yield curves coincide with each other.

7. The (per-period) holding return for a coupon bond (C,F, T ) for theperiod (t, T ) is defined as

C

∑T−t−1j=1 ΠT−t−j−1

k=0 [1 + Yt+j+k(1)] + (F + C)

Pt

1

T−t − 1.

In other words, it is the per-period return on the date-t buy-and-holdstrategy, with all coupon payments being continually re-invested in theone-period bonds (or equivalently, in a money market account) till dateT . It is obvious that the holding return is random, so that the buy-and-hold strategy is a risky strategy. This should be contrasted with thedate-t buy-and-hold strategy applied to a zero-coupon bond maturingat date T . The latter is a riskless strategy, and its holding return isthe non-random constant

[1

B(t, T − t)]

1T−t − 1 = [1 + Yt(T − t)]T−t

1T−t − 1 = Yt(T − t).

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Note that the holding return for a coupon bond (C,F, T ) for the period(t, T ) gets higher (respectively, lower) if Yt+j(·) shifts up (respectively,down) for all j = 1, 2, · · · , T − t− 1.

In particular, if for all j = 1, 2, · · · , T − t − 1, Yt+j(·) = xt, where xt

is the date-t YTM for the coupon bond (C,F, T ), then the per-periodholding return for the coupon bond for the period (t, T ) is exactly xt.To see this, note that

C

∑T−t−1j=1 [1 + xt]

T−t−j + (F + C)

Pt

1

T−t − 1

= C

∑T−t−1j=1 [1 + xt]

T−t−j + (F + C)

F · (1 + xt)t−T + C∑T−t

k=1(1 + xt)−k

1T−t − 1

= 11

(1+xt)T−t

[C

∑T−t−1j=1 [1 + xt]

T−t−j + (F + C)

C∑T−t−1

j=1 [1 + xt]T−t−j + (F + C)]

1T−t − 1

= 11

(1+xt)T−t

[1]1

T−t − 1 = xt.

Hence if for all j = 1, 2, · · · , T − t−1, Yt+j(·) shifts above (respectively,below) xt, then the per-period holding return is greater (respectively,less) than xt.

8. We see in the last section that when the future interest rates Yt+j(·)change, they affect the returns on the re-investments of coupon pay-ments. There is also a countervailing effect: they also affect the date-(t+ j) price of the coupon bond. Recall that

Pt+j = F · B(t+ j, T − t− j) + CT−t−j∑k=1

B(t+ j, k),

where recall also that

B(t+ j, k) =1

[1 + Yt+j(k)]k,

so that Pt+j becomes lower when Yt+j(·) shifts up. Thus interest raterisk (that future yield curves Yt+j(·) are random from date-t perspec-tive) has two effects on the profits of bond trading: when Yt+j(·) shifts

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up, reinvestments of coupon payments make more profits, but the date-(t+ j) bond price goes down.

9. To take a look at how the price of a coupon bond may change when theyield curve moves over time, we can at first investigate the simplest sit-uation where the yield curve never moves, and it is flat at the constantlevel x. Consider for example at date 0 the coupon bond (C,F, T ) withT = 3. We know that its date-0 price is

P0 =C

(1 + x)+

C

(1 + x)2+

C + F

(1 + x)3,

and if x = CF, then

P0 = F.

Similarly, we have

F = P1 =C

1 + x+

C + F

(1 + x)2,

if x = CF, and hence in this special case we have

P1 = P0,

so that the bond price does not change as time goes by! This is notsurprising: if at date 0 you spend P0 in a one-period time deposit,you will get P0(1 + x) = P0 + P0x at date 1; and if you hold 1 unitof the above coupon bond from date 0 to date 1, and then sell thebond at date 1, you will get P1 + C at date 1. To rule out arbitrageopportunities, we must have

P0 + P0x = P1 + C,

and given that P0 = F and x = CF, we must have P1 = F also.

As you can easily verify, in case x > CF, then the bond is sold at date

0 at a discount, and we have

P0 < P1 < F ;

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and in case x < CF, then the bond is sold at date 0 at a premium, and

we haveP0 > P1 > F.

Our conclusion is that, if the yield curve is flat and will never move,then the bond price will rise (respectively, fall) over time and convergetoward the par value as time goes by, if the coupon rate C

Fis less

(respectively, greater) than the exogenously given discount rate x.

What if the yield curve never moves, but it is non-flat? Again, considerat date 0 the coupon bond (C,F, T ) with T = 3. We know that ingeneral the bond’s date-0 price is

P0 =C

(1 + Y0(1))+

C

(1 + Y0(2))2+

C + F

(1 + Y0(3))3,

and its date-1 price becomes

P1 =C

(1 + Y1(1))+

C + F

(1 + Y1(2))2.

Since we have assumed that the yield curve never moves, for all t,Y0(t) = Y1(t). Now, if Y0(·) is upward sloping, in the sense that

Y0(1) < Y0(2) < Y0(3),

then we have

C

(1 + Y0(1))+

C

(1 + Y0(2))2+

C + F

(1 + Y0(3))3= P0 < P1 =

C

(1 + Y0(1))+

C + F

(1 + Y0(2))2,

except in the case where C is very large (where the bond is sold at deeppremium at date 0). Conversely, if Y0(·) is downward sloping, in thesense that

Y0(1) > Y0(2) > Y0(3),

then we have

C

(1 + Y0(1))+

C

(1 + Y0(2))2+

C + F

(1 + Y0(3))3= P0 > P1 =

C

(1 + Y0(1))+

C + F

(1 + Y0(2))2,

except in the case where C is very small (where the bond is sold atdeep discount at date 0).

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To sum up, even in the case where the yield curve never moves, theprice of a coupon bond may still change over time. We call it the ma-turity effect. There are two components in the maturity effect. First,if the coupon rate is set very high (respectively, low) relative to theexogenously given yield curve, then the bond price must be sold ata high premium (respectively, discount) at the beginning, and as thecoupon payments are made to the bondholder date after date, the newbond price will fall (respectively, rise) gradually, and ultimately con-verge toward the face value of the bond. Second, if the yield curveis upward (respectively, downward) sloping, then as time goes by, thediscount rates applied to the unpaid coupons and the par value fall (re-spectively, rise) steadily, so that the bond price (which is the presentvalue of the unpaid coupons and the final face value computed usingthose discount rates) rises (respectively, falls) over time.

10. The preceding section describes what would happen if the yield curvenever moves. In reality, the yield curve moves up and down over time ina stochastic fashion. We have emphasized that this stochastic changein the yield curve (or, the interest rate risk, simply) has two opposingeffects on the profitability of bond trading. First, it affects the profitsof re-investing the coupons received at earlier dates, thereby affectingthe return on the buy-and-hold strategy (or, simply, the holding re-turn). Second, it changes the discount rates for the un-paid couponsand the final par value, thereby changing the bond price and resultingin capital gains or losses when a bondholder wishes to sell the bondbefore maturity date.

For various reasons, an investor may want to remove the interest raterisk in bond trading. If the yield curve is random but always flat, thenthe date-t bond price depends on the date-t interest rate xt via

Pt(xt) =T−t∑j=1

C

(1 + xt)j+

F

(1 + xt)T−t.

Note that the function Pt(·) is smooth (meaning that it possesses con-tinuous derivatives at all orders as long as the interest rate xt > 0),

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and by Taylor’s expansion formula we have

Pt(x′)− Pt(x) = P ′

t(x)(x′ − x) +

1

2!P ′′t (x)(x

′ − x)2 +1

3!P ′′′t (z)(x′ − x)3,

where z is some number lying between x and x′. Note that Pt(x′)−Pt(x)

is the difference in the date-t bond price resulting from the interest ratext changing from the x to x′. Hence Pt(x

′) − Pt(x) is the capital gainor loss in holding the bond at date t for one second, when the interestrate changes suddenly from the level x to x′.

To remove, or at least minimize the interest rate risk from bond trading,we must first measure how much the bond price may change followinga sudden change in the interest rate. The above Taylor’s expansiontells us that, if |x′−x| is small enough, so that we can ignore the termsinvolving (x′ − x)2, (x′ − x)3, and so on, then the larger |P ′

t(x)| is, themore the bond price will change as a response to a sudden change inthe interest rate.

Formally, we call the point elasticity of Pt to (1+xt) as the (Macaulay’s)duration, and as its definition suggests, a bond with a larger dura-tion is more vulnerable to the interest rate risk. Denote the date-t(Macaulay’s) duration of the coupon bond (C,F, T ) at the current in-terest rate xt by

Dt = −d log(Pt(xt))

d log(1 + xt).

It is straightforward to verify that6

D0 = −d log(P0(x0))

d log(1 + x0)=

1

P0

T · F(1 + x0)T

+T∑t=1

t · C(1 + x0)t

=T∑t=1

wt · t,

6As an exercise, show that if F = 1, then

D0 =1 + x0

x0− T (C − x0) + 1 + x0

C[(1 + x0)T − 1] + x0.

This implies that as T tends to infinity, D0 tends to 1 + 1x0.

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where the weights

0 ≤ wt =

C

(1+x0)t

P0, t = 1, 2, · · · , T − 1;

F+C

(1+x0)T

P0, t = T.

Note that by the definitions of P0 and of x0,

T∑t=1

wt = 1,

and hence D0 ≤ T gives an average payback period when the buy-and-hold strategy is adopted at date 0.7 More generally, Dt is a weightedaverage of τ = 1, 2, 3, · · · , T − t, with the weight wτ for τ being theratio of the present value of the date-(t+ τ) cash flow generated by thebuy-and-hold strategy adopted at date t, to Pt. This is no coincidence,for as we mentioned, the longer the maturity the more sensitive thebond price is to changes in interest rates. In particular, if C = 0,then D0 = T ! That is, with F and T fixed, the price of a couponbond is most sensitive to changes in interest rates when it is in fact azero-coupon bond. For this reason, zero-coupon bonds have the highestdurations (and hence their prices are most vulnerable to interest raterisk).

11. The duration of a portfolio of bonds is a value-weighted average of thedurations of the ingredient bonds. For i = 1, 2, · · · , n, let the date-tprice of bond i be P i

t (xt), as a function of the date-t YTM xt, wherewe assume a flat yield curve. Consider a portfolio of these n bonds,with the portfolio weight of bond i being wi. The date-t value of thisportfolio is

n∑i=1

wiPit (xt),

7Associated with D0, the following expression is referred to as the modified Macaulay’sduration:

Dmod0 ≡ −dP0

dx0· 1

P0=

D0

1 + x0.

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so that the duration of this portfolio is

Dt ≡d log(

∑ni=1wiP

it (xt))

d log(1 + xt)=

n∑i=1

λiDit,

where Dit is the date-t duration of portfolio i, and

λi =wiP

it (xt)∑n

j=1 wjPjt (xt)

.

Hence the date-t duration Dt of the portfolio of bonds is a weightedaverage of the durations of ingredient bonds. This fact shows how wemight use a set of bonds with various durations to form a portfolio ofbonds with a desired duration.

12. Why would an investor want to hold a portfolio of bonds with some pre-specified duration? One possibility is that at date 0, say, the investorknows that she needs to generate a pre-specified amount of cash at afuture date D < T . Suppose that the date-0 yield curve is flat at x0,and that the yield curve may make a one-time parallel random shift oneinstant after date 0 from x0 to x and then stay at x forever. We claimthat if the shift |x − x0| is infinitesimal, then continually re-investingall the coupon payments received before date D at the new YTM x tilldate D and then selling the bond at date D will generate at date D anamount of cash L(x) which is independent of the realization of x!

To see this, simply define ϵ ≡ x − x0 and verify that by the definitionof the duration D, we have8

d

dϵL(x0 + ϵ)|ϵ=0 = 0,

8Note thatL(x) = P0(x)(1 + x)D,

so thatL′(x)dx = dL(x) = P0(x)d[(1 + x)D] + d[P0(x)](1 + x)D

= P0(x)D(1 + x)D−1dx+ (1 + x)DdP0(x)

= P0(x)D(1 + x)D−1d(1 + x) + (1 + x)DdP0(x)

= (1 + x)D[P0(x) ·d(1 + x)

1 + x·D] + dP0(x)

15

where

L(x) ≡T∑t=1

C(1 + x)D−t + F (1 + x)D−T .

What happens here is that, if x0 rises slightly in a second and then staysthere, then the date-D bond price will fall, but the coupon paymentsreceived before date D can be re-invested at a higher rate of return,which exactly offsets the reduction in the date-D bond price caused bythe increase in x0; and if x0 falls slightly in a second and then staysthere, then the date-D bond price will rise, but the coupon paymentsreceived before date D can be re-invested only at a lower rate of return,which exactly offsets the increase in the date-D bond price caused bythe fall in x0. Hence, for tiny one-time changes in x0, the re-investing-coupon-payments-and-selling-at-date-D strategy does generate a cashflow which is fully immuned from the interest rate risk.

We conclude that if you have a sure liability Z due at date D, then youcan adopt the immunization strategy of holding a portfolio of bondswith duration equal to D, which, if the interest rate risk does presentitself as we described (to shift slightly in a parallel fashion and to shiftonly once), will allow you to generate a sure amount of cash meetingthe liability due at date D.9

13. In reality, the yield curve is always non-flat, and in that case we canadopt the following (date-0) generalized Macaulay’s duration

Dg0 ≡

1

P0

− d

dϵ[T∑

s=1

C

(1 + Y0(s) + ϵ)s+

F

(1 + Y0(T ) + ϵ)T]|ϵ=0

=1

P0

T∑t=1

tC

(1 + Y0(t))t+1+

TF

(1 + Y0(T ))T+1.

= (1 + x)D[P0(x) ·d(1 + x)

1 + x· (−

dP0(x)P0(x)

d(1+x)1+x

)] + dP0(x)

= (1 + x)D−dP0(x) + dP0(x) = 0.

9If the zero-coupon bond B(0, D) is available for trading, then holding Z units ofB(0, D) will serve the purpose.

16

Again, the generalized duration measures the percentage change in thebond price with respect to an infinitesimal change in ϵ, which representsa one-time parallel shift in the yield curve in the next instant.

14. The duration technique has been adopted by the savings and loan insti-tutions to try to remove the impact of interest rate changes on their networths. Recall that these institutions’s assets are mainly composed ofloans, and their liabilities are mainly composed of deposits, and thosetwo can be regarded as fixed-income securities.10 Let the date-t valuesof an S& L institution’s assets and liabilities be denoted by respectivelyAt(xt) and Lt(xt), with the latter two’s date-t durations being respec-tively DA

t and DLt , and the institution’s date-t equity, or net worth, be

denoted by Et (which is the difference between the At(xt) and Lt(xt)).

Now, imagine that at date t, the interest rate suddenly changes from xto x′, and then it stays at x′ forever. How much does this change theS& L institution’s net worth at date t?

Again, if we assume that |x′−x| is small enough, so that we can ignorethe terms involving (x′−x)2, (x′−x)3, and so on, then approximately,

Et(x′)−Et(x) = [At(x

′)−At(x)]−[Lt(x′)−Lt(x)] ∼ [AtD

At −LtD

Lt ]x′ − x

1 + x.

Hence, if we want Et to be independent of the new x′, we need to set

AtDAt = LtD

Lt ,

which is called a duration match.11

10Recall that we have assumed no credit risks, which is very unrealistic. We shall alsoassume that the interest rates applied to loans and deposits are the same, which is equallyunrealistic. To apply these theories in practice, the formulae to appear below must bemodified.

11Here is another proof, perhaps more illuminating. Note that E = A − L, and hencedE = dA− dL. Note that

dA = A · d(1 + Y TM)

1 + Y TM·DA,

dL = L · d(1 + Y TM)

1 + Y TM·DL.

17

Now, the next question is, given At and Lt, how to setDAt (andDL

t also)equal to a particular value, say 6 years? Recall that the institution’sassets are loans, and each loan can be regarded as a coupon bond. Wehave discussed in a preceding section regarding how to adjust the loanportfolio so that the entire portfolio of coupon bonds has a desiredduration, say 6 years.

15. Recall the Taylor’s expansion formula

Pt(x′)− Pt(x) = P ′

t(x)(x′ − x) +

1

2!P ′′t (x)(x

′ − x)2 +1

3!P ′′′t (z)(x′ − x)3,

where z is some number lying between x and x′. As should be clearright now, if |x′−x|2 is not so small to be ignored, but |x′−x|n is smallenough and can be safely ignored for all n ≥ 3, then to account for theimpact of the change in the date-t interest rate on Pt, we must takeinto account both P ′

t(x)(x′−x) and 1

2!P ′′t (x)(x

′−x)2. The second-orderterm is always positive, whether x′ > x or x′ < x, and is referred to asconvexity, indicating the fact that Pt(·) is a convex function. Formally,define

0 < V ≡ 1

2

d2P0

dx20

· 1

P0

.

The duration and the convexity are prevalent instruments people use tohedge the interest-rate risks pertaining to the trading of fixed-incomesecurities. For a thorough discussion of interest rate hedging using bothduration and convexity, see for example de La Grandville, 2001, BondPricing and Portfolio Analysis, Cambridage: MIT Press.

16. Continuous-time Finance has formalized a procedure to hedge the in-terest rate risk, in a context which is much more realistic than theabove environment (where the yield curve is assumed to be either flat,or to move in a parallel fashion). For example, an interest-rate-sensitivecontingent claim (including any coupon bond) may have a date-t price

We want dE = 0; that is, we want the equity value to be immuned from the interest raterisk. Note that

0 = dE =d(1 + Y TM)

1 + Y TM· [ADA − LDL],

so that we need ADA = LDL.

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P (t, y1t, y2t, · · · , ynt), where (y1t, y2t, · · · , ynt) is an n-dimensional statevariable that completely describes the economy at date t. We call thisan n-factor model. To give a simple example, let n = 1, and in thisone-factor model,

y1t = Yt(dt) ≡ r(t);

that is, the only state variable is the interest rate applied to a loanmade at time t and repaid at time t + dt. We call r(t) the short rateprocess, which corresponds to the short rates r0,1, r1,2(E) and r1,2(E

c)in Lecture 5.

In continuous-time finance we usually assume that

dr(t) ≡ r(t+ dt)− r(t) = µ(t)dt+ σ(t)dB(t),

where dB(t), satisfying a set of very interesting behavioral restrictions,is a normal random variable with mean and variance equal to respec-tively 0 and dt, and µ(t) and σ(t) are themselves some stochastic ornon-stochastic processes. We call B(t) a (standard) Brownian motion,and dB(t) its stochastic differential. Having descrived how the statevariable r(t) may evolve over time, one can decide how the contingentclaim’s price may change over time. This again bears on the Taylor’sexpansion formula (with two arguments):

dP (t, r(t)) =∂P

∂tdt+

∂P

∂r(t)[dr(t)] +

1

2!

∂2P

(∂r(t))2[dr(t)]2 + · · · ,

where [dr(t)]2 composed of terms such like dtdt, dtdB(t), and dB(t)dB(t),and as a convention (which we call Ito’s Lemma), only dB(t)dB(t) isnon-zero, and actually dB(t)dB(t) equals dt. Ito’s Lemma shows thatall the terms in the Taylor’s expansion that we did not write downexplicitly are actually equal to zero.

This last equation shows how, say, a coupon bond’s price may changeover time: the first term on the righ side is the maturity effect thatwe mentioned above, and the second and third terms capture the du-ration effect and the convexity effect. In an advanced course on assetpricing or financial engineering, you will learn how to create a dynamichedging strategy to remove the latter two effects by continuously rebal-ancing your portfolio. Theoretically, those dynamic strategies require

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you to trade in each and every instant in time. In reality, transactioncosts are a major concern for carrying out such a dynamic hedgingstrategy, and hence a modified strategy that requires frequent but notcontinuous trading may actually be adopted. Such a dynamic hedgingstrategy typically works better than the above duration and convexitytechniques.

17. Now we give a series of examples.

Example 2 Consider a three-date (dates 0, 1 and 2) economy withperfect financial markets. There are four possible states (Ω = ω1, ω2, ω3, ω4).At date 0, investors only know that the true state is an element of Ω,and the true state will be revealed perfectly at date 2. At date 1, the oc-currence or non-occurrence of the event E = ω1, ω2 becomes publiclyknown. This is the standard event tree that we considered in Lecture5. We assume that markets are dynamically complete. The followingtable gives the price process of asset 1, denoted by p1, and the forwardprice process for the delivery of one unit of asset 1 at date 2, which isdenoted by G.

Asset price/(Date,Event) (0,Ω) (1, E) (1, Ec) (2, ω1) (2, ω2) (2, ω3) (2, ω4)p1 100 110 90 150 114 134 100G 624

5132 117 150 114 134 100

At date 0, two default-free coupon bonds, referred to as X and Y, aretraded, and their data are summarized in the following table (where thedate-0 bond prices are obtained after the date-0 coupon payments aremade):

bond maturity coupon payment face value date-0 priceX date 1 50 1000 xY date 2 200 1000 y

(i) Compute x and y.(ii) Mr. B is endowed with 1 million dollars (a sure income!) at date2. He only wants to consume at date 1, and he wants to consume the

20

same amount in event E and in event Ec. Suppose that he is allowedto trade the above two bonds X and Y only, and he is allowed to tradeonly at date 0. At date 0, how many units of bond X should Mr. B buyor sell, and how many units of bond Y should Mr. B buy or sell? Howmuch can Mr. B consume at date 1?(iii) At date 0, what is the futures price for the delivery of one unit ofasset 1 at date 2?

Solution. From the price processes p1 and G, we can recover the priceprocess B for the zero-coupon bond that will mature at date 2 withface value equal to 1 dollar. In fact, we have

B(0) =p1(0)

G(0)=

125

156, B(1, E) =

p1(1, E)

G(1, E)=

5

6, B(1, Ec) =

p1(1, Ec)

G(1, Ec)=

10

13,

B(2, ω1) = B(2, ω2) = B(2, ω3) = B(2, ω4) = 1.

Now, since asset 1 and the zero-coupon bond are linearly independent,we can use their price processes to recover the short-rate process andthe martingale probabilities for the 4 states. Following the standardprocedure outlined in Lecture 5A, we can obtain the short rate process

r0,1 = 0, r1,2(E) = 0.2, r1,2(Ec) = 0.3,

and the martingale probabilities

π(ω1) = π(ω2) = π(ω3) = π(ω4) =1

4.

Now we can start solving parts (i)-(iii).

First, for part (i), we have

x =1050

1 + r0,1= 1050, y =

200

1 + r0,1+ 1200B(0) = 1161.54.

Next, consider part (ii). Consider selling 1,000,0001,200

= 833.33 units of

bond Y and buying 1,000,000y1,200x

= 921.86 units of bond X. Apparently,

21

this will generate no cash inflow or outflow at date 0. The date-2payoff generated by this strategy is

−1, 000, 000

1, 200× 1, 200 = −1, 000, 000;

that is, a 1-million-dollar cash outflow. This cash outflow exactly offsetsMr. B’s date-2 income, which is also 1 million dollars, leaving no cashat date 2. What about date 1? This strategy generates a date-1 payoffequal to

−1, 000, 000

1, 200× 200 +

1, 000, 000y

1, 200x× 1050 = 801, 283.3,

which is the amount that Mr. B can consume at date 1, regardless ofwhether event E or event Ec occurs. This finishes part (ii).

Finally, consider part (iii). It is easily verified that the date-0 futuresprice

H(0) =1

2[H(1, E) +H(1, Ec)] =

1

2[G(1, E) +G(1, Ec)] = 124.5.

18. Example 3 At date 0, three coupon bonds are traded, and their dataare summarized in the following table (where the date-0 bond prices areobtained after the date-0 coupon payments are made):

maturity coupon payment face value date-0 pricedate 1 100 1000 1045date 2 80 1000 1048date 3 40 1000 906

Mr. A will receive no income except at date 2, and his date-2 incomeis one million dollars. He wants to consume a sure amount (only) atdate 3. How much can Mr. A consume at date 3 if he is allowed totrade the above three coupon bonds (only) at date 0?

Solution. Let B(0, t) be the date-0 price of a pure discount bondmaturing at date t with face value equal to one dollar. From the data

22

given above, we have1, 100×B(0, 1) = 1, 045;80×B(0, 1) + 1, 080×B(0, 2) = 1, 048;40×B(0, 1) + 40×B(0, 2) + 1, 040×B(0, 3) = 906.

Solving, we have

B(0, 1) = 0.95, B(0, 2) = 0.9, B(0, 3) = 0.8.

Mr. A should short sell one million units of bond B(0, 2) (and useits date-2 income to clear this short position at date 2), and put theproceeds 1, 000, 000 × B(0, 2) in bond B(0, 3). Thus the sure amountMr. A can consume at date 3 is

1, 000, 000×B(0, 2)

B(0, 3)=

1, 000, 000× 0.9

0.8= 1, 125, 000.

19. Example 4 At date 0, three coupon bonds are traded, and their dataare summarized in the following table (where the date-0 bond prices areobtained after the date-0 coupon payments are made):

maturity coupon payment face value date-0 pricedate 1 50 1000 997.5date 2 200 1000 1270date 3 100 1000 1120

Mr. A will receive no income except at date 3, and his date-3 incomeis one million dollars. He wants to consume a sure and equal amountover date 1, date 2 and date 3. How much can Mr. A consume ateach of these three dates, if he is only allowed to trade the above threecoupon bonds at date 0?

Solution. We first recover the prices of pure discount bonds with facevalue equal to one dollar. It is easy to get

B(0, 1) = 0.95, B(0, 2) = 0.9, B(0, 3) = 0.85.

(To verify: we have

100×B(0, 1) + 100×B(0, 2) + (1000 + 100)×B(0, 3) = 1120,

23

200×B(0, 2) + (1000 + 200)×B(0, 1) = 1270,

and(50 + 1000)×B(0, 1) = 997.5.)

Let the unknown consumption of Mr. A at each of the three dates bex. We must have

x× [B(0, 1) +B(0, 2) +B(0, 3)] = 1, 000, 000×B(0, 3).

Solving, we have approximately

x = 314, 815.

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