fixed income solution ch3

7/29/2019 fixed income solution ch3

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15Problems and Solutions

3 CHAPTER 3 Problems

Exercise 3.1 We consider three zero-coupon bonds (strips) with the following features:

Bond Maturity (years) Price

Bond 1 1 96.43Bond 2 2 92.47

Bond 3 3 87.97

Each strip delivers $100 at maturity.

1. Extract the zero-coupon yield curve from the bond prices.

2. We anticipate a rate increase in one year so the prices of strips with residual

maturity 1 year, 2 years and 3 years are respectively 95.89, 90.97 and 84.23.

What is the zero-coupon yield curve anticipated in one year?

Solution 3.1 1. The 1-year zero-coupon rate denoted by R(0, 1) is equal to 3.702%

R(0, 1) =100

96.43 1 = 3.702%

The 2-year zero-coupon rate denoted by R(0, 2) is equal to 3.992%

R(0, 2) =

100

92.47

1/2 1 = 3.992%

The 3-year zero-coupon rate denoted by R(0, 3) is equal to 4.365%

R(0, 2) =

100

87.97

1/3 1 = 4.365%

2. The 1-year, 2-year and 3-year zero-coupon rates become respectively 4.286%,4.846% and 5.887%.

Exercise 3.3 We consider the following decreasing zero-coupon yield curve:

Maturity (years) R(0, t ) (%) Maturity (years) R(0, t) (%)

1 7.000 6 6.250

2 6.800 7 6.200

3 6.620 8 6.160

4 6.460 9 6.125

5 6.330 10 6.100

where R(0, t ) is the zero-coupon rate at date 0 with maturity t.

1. Compute the par yield curve.

2. Compute the forward yield curve in one year.

3. Draw the three curves on the same graph. What can you say about their relative

position?


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Solution 3.3 1. Recall that the par yield c(n) for maturity n is given by the formula

c(n) =1 1

(1+R(0,n))nni=1

1(1+R(0,i))i

Using this equation, we obtain the following par yields:

Maturity (years) c(n) (%) Maturity (years) c(n) (%)

1 7.000 6 6.293

2 6.807 7 6.246

3 6.636 8 6.209

4 6.487 9 6.177

5 6.367 10 6.154

2. Recall that F (0, x , y x), the forward rate as seen from date t = 0, starting at

date t = x, and with residual maturity y x is defined as

F (0, x , y x) (1 + R(0,y))y

(1 + R(0,x))x 1yx

1

Using the previous equation, we obtain the forward yield curve in one year

Maturity (years) F (0, 1, n) (%) Maturity (years) F (0, 1, n) (%)

1 6.600 6 6.067

2 6.431 7 6.041

3 6.281 8 6.016

4 6.163 9 6.000

5 6.101

3. The graph of the three curves shows that the forward yield curve is below the

zero-coupon yield curve, which is below the par yield curve. This is always thecase when the par yield curve is decreasing.

5.75

6.00

6.25

6.50

6.75

7.00

7.25

1 2 3 4 5 6 7 8 9 10

Maturity

Yield(%)

Par yield curve

Zero-coupon yield curve

Forward yield curve


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Exercise 3.8 When the current par yield curve is increasing (respectively, decreasing), the cur-

rent zero-coupon rate curve is above (respectively, below) it, so as to offset the fact

that the sum of the coupons discounted at the coupon rate is inferior (respectively,

superior) to the sum of the coupons discounted at the zero-coupon rate. Give a

proof of this assertion.

Solution 3.8 Let us denote by c(i) , the par yield with maturity i and by R(0, i), the zero-coupon

rate with maturity i.

Let us assume for k < n that c(n) > c(k)

At the first rank, we have R(0, 1) = c(1)

At the second rank,

c(2)

1 + c(2)+

1 + c(2)

(1 + c(2))2=

c(2)

1 + R(0, 1)+

1 + c(2)

[(1 + R(0, 2)]2

Let us do a limited development at the first order of this last expression. Then

c(2).(1 c(2)) + (1 + c(2)).(1 2c(2)) = c(2).(1 R(0, 1))

+ (1 + c(2)).(1 2R(0, 2))

R(0, 2) c(2) =1

2.

c(2)

1 + c(2).(c(2) c(1))

As c(2) > c(1), R(0, 2) > c(2).

The proposition is true at the first and second ranks. Let us assume that it is

true at rank n 1 and let us prove it at rank n.

nk=1

c(n)

(1 + c(n))k+

1

(1 + c(n))n=

nk=1

c(n)

(1 + R(0,k))k+

1

(1 + R(0, n))n

Let us do again a limited development at the first order. Then

nk=1

c(n).(1 kc(n)) + 1 nc(n) =

nk=1

c(n).(1 kR(0,k)) + 1 nR(0, n)

After simplification

R(0, n) c(n) =1

n.

c(n)

1 + c(n).

n

k=1

k.(c(n) c(k)) +c(n 1)

1 + c(n 1)

n1k=1

c(k)

as c(n1)1+c(n1)n1

k=1 c(k) can be considered negligible sincec(n1)

1+c(n1) is close to

zero, we can make the following approximation:

R(0, n) c(n) = 1n

. c(n)1 + c(n)

.

nk=1

k.(c(n) c(k))

As c(n) > c(k), we obtain R(0, n) > c(n), which proves the assertion.

Exercise 3.11 At date t = 0, we observe the following zero-coupon rates in the market:


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Maturity Zero-Coupon Maturity Zero-Coupon

(years) Rate (%) (years) Rate (%)

1 5.00 4 6.80

2 6.00 5 7.00

3 6.50

1. What are the 1-year maturity forward rates implied by the current term structure?

2. Over a long period, we observe the mean spreads between 1-year maturity for-

ward rates and 1-year maturity realized rates in the future. We find the following

liquidity premiums:

L2 = 0.1%

L3 = 0.175%

L4 = 0.225%

L5 = 0.250%

Taking into account these liquidity premiums, what are the 1-year maturity

future rates expected by the market?

Solution 3.11 1. 1-year maturity forward rates are given by the following formula:

[1 + R(0, T )]T = [1 + R(0, T 1)]T1 . [1 + F (0, T 1, 1)]

where R(0, T ) is the zero-coupon rate at date t = 0 with T-year maturity

and F (0, T 1, 1) is the 1-year maturity forward rate observed at date t = 0,

starting at date t = T 1 and maturing one year later.

F (0, 4, 1) is obtained by solving the following equation:

F (0, 4, 1) =(1 + 7%)5

(1+

6.8%)4

1 = 7.804%

Using the same equation, we obtain

Forward Rates

F (0, 1, 1) 7.009%

F (0, 2, 1) 7.507%

F (0, 3, 1) 7.705%

2. 1-year maturity future rates expected by the market are given by the following

formula:

[1 + R(0, T )]T = [1 + R(0, T 1)]T1.[1 + Fa (0, T 1, 1) + LT]

where Fa (0, T 1, 1) is the 1-year maturity future rate expected by the market

at date t = 0, starting at date t = T 1 and finishing one year later.

Using the last equation, we find the relation between the forward rate and

the future rate expected by the market

Fa (0, T 1, 1) = F (0, T 1, 1) LT


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We finally obtain

Expected Future Rates

Fa (0, 1, 1) 6.909%

Fa (0, 2, 1) 7.332%

Fa (0

,3

,1

)7

.480%Fa (0, 4, 1) 7.554%

Exercise 3.12 Monetary policy and long-term interest rates

Consider an investor with a 4-year investment horizon. The short-term (long-

term respectively) yield is taken as the 1-year (4-year respectively) yield. The

medium-term yields are taken as the 2-year and 3-year yields. We assume, fur-

thermore, that the assumptions of the pure expectations theory are valid.

For each of the following five scenarios, determine the spot-yield curve at

date t = 1. The yield curve is supposed to be initially flat at the level of 4%, at

date t = 0.

(a) Investors do not expect any Central Bank rate increase over four years.(b) The Central Bank increases its prime rate, leading the short-term rate

from 4 to 5%. Investors do not expect any other increase over four years.

(c) The Central Bank increases its prime rate, leading the short-term rate from

4 to 5%. Investors expect another short-term rate increase by 1% at the beginning

of the second year, then no other increase over the last two years.

(d) The Central Bank increases its prime rate, leading the short-term rate

from 4 to 5%. Nevertheless, investors expect a short-term rate decrease by 1% at

the beginning of the second year, then no other change over the last two years.

(e) The Central Bank increases its prime rate, leading the short-term rate

from 4 to 5%. Nevertheless, investors expect a short-term rate decrease by 1%

each year, over the following three years.What conclusions do you draw from that as regards the relationship existing

between monetary policy and interest rates?

Solution 3.12 Let us denote Fa (1,n,m), the m-year maturity future rate anticipated by the mar-

ket at date t = 1 and starting at date t = n, and R(1, n) the n-year maturity

zero-coupon rate at date t = 1.

In each scenario, we have

Scenario a Scenario b Scenario c Scenario d Scenario e% % % % %

R(1, 1) 4.00 5.00 5.00 5.00 5.00

Fa (1, 2, 1) 4.00 5.00 6.00 4.00 4.00

Fa (1, 2, 2) 4.00 5.00 6.00 4.00 3.00

Fa (1, 2, 3) 4.00 5.00 6.00 4.00 2.00

Using the following equation:

1 + R(t, n) = [(1 + R(t, 1))(1 + Fa (t,t + 1, 1))(1 + Fa (t,t + 2, 1) ) . . .

(1 + Fa (t,t + n 1, 1))]1/n


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we find the spot zero-coupon yield curve in each scenario.

Scenario a Scenario b Scenario c Scenario d Scenario e

% % % % %

R(1, 1) 4.00 5.00 5.00 5.00 5.00

R(1, 2) 4.00 5.00 5.50 4.50 4.50R(1, 3) 4.00 5.00 5.67 4.33 4.00

R(1, 4) 4.00 5.00 5.75 4.25 3.49

In the framework of the pure expectations theory, monetary policy affects

long-term rates by directly impacting spot and forward short-term rates, which

are supposed to be equal to market short-term rate expectations. But what about

these expectations? The purpose of the exercise is to show that market short-

term rate expectations play a determining role in the response of the yield curve

to monetary policy. More meaningful than the Central Bank action itself is the

way market participants interpret this action. Is it a temporary action or rather the

beginning of a series of similar actions. . . ? We can draw three conclusions fromthe exercise.

First, the direction taken by interest rates compared with that of the Cen-

tral Bank prime rate depends on the likelihood, perceived by the market, that the

Central Bank will question its action in the future through reversing its stance.

Under the (b) and (c) scenarios, the Central bank action is perceived to either fur-

ther increase its prime rate or leave things as they are. Consequently, long-term

rates increase following the increase in the prime rate. Under the (d) scenario, the

Central Bank is expected to exactly offset its increasing action in the future. Nev-

ertheless, its action on short-term rates still remains positive over the period. As

a result, long-term interest rates still increase. In contrast, under the (e) scenario,

the Central Bank is expected to completely reverse its stance through a decreas-

ing action in the future, that more than offsets its initial action. Consequently,

long-term interest rates decrease.

Second, the magnitude of the response of long-term rates to monetary pol-

icy depends on the degree of monetary policy persistence that is expected by the

market. Under the (b) and (c) scenarios, the Central Bank action is viewed as rel-

atively persistent. Consequently, the long-term interest-rate change either reflects

the instantaneous change in the prime rate or exceeds it. Under the (d) scenario, as

the Central Bank action is perceived as temporary, the change in long-term rates

is smaller than the change in the prime rate.

Third, the reaction of long-term rates to monetary policy is more volatilethan that of short-term rates. That is, the significance of the impact of market

expectations on interest rates increases with the maturity of interest rates. These

expectations only play a very small role on short-term rates. As illustrated by

the exercise, the variation margin of the 2-year interest rate following a 100-bps

increase of the Central Bank prime rate is contained between 50 and 150 bps, while

the variation margin of the 4-year interest rate is more volatile (between 50 bp

and +175 bps).


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Exercise 3.13 Explain the basic difference that exists between the preferred habitat theory and

the segmentation theory.

Solution 3.13 In the segmentation theory, investors are supposed to be 100% risk-averse. So

risk premia are infinite. It is as if their investment habitat were strictly con-

strained, exclusive.In the preferred habitat theory, investors are not supposed to be 100% risk

averse. So, there exists a certain level of risk premia from which they are ready

to change their habitual investment maturity. Their investment habitat is, in this

case, not exclusive.


Exercise 4.1 At date t = 0, we consider five bonds with the following features:

Annual

Coupon Maturity PriceBond 1 6 1 year P10 = 103

Bond 2 5 2 years P20 = 102

Bond 3 4 3 years P30 = 100

Bond 4 6 4 years P40 = 104

Bond 5 5 5 years P50 = 99

Derive the zero-coupon curve until the 5-year maturity.

Solution 4.1 Using the no-arbitrage relationship, we obtain the following equations for the five

bond prices:

103 = 106B(0, 1)

102 = 5B(0, 1) + 105B(0, 2)

100 = 4B(0, 1) + 4B(0, 2) + 104B(0, 3)

104 = 6B(0, 1) + 6B(0, 2) + 6B(0, 3) + 106B(0, 4)

99 = 5B(0, 1) + 5B(0, 2) + 5B(0, 3) + 5B(0, 4) + 105B(0, 5)

which can be expressed in a matrix form as

103

102

100

104

99

=

106

5 105

4 4 104

6 6 6 106

5 5 5 5 105

B(0, 1)

B(0, 2)

B(0, 3)

B(0, 4)

B(0, 5)

We get the following discount factors:

B(0, 1)

B(0, 2)

B(0, 3)

B(0, 4)

B(0, 5)

=

0.97170

0.92516

0.88858

0.82347

0.77100


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and we find the zero-coupon rates

R(0, 1) = 2.912%

R(0, 2) = 3.966%

R(0, 3) = 4.016%

R(0, 4) = 4.976%

R(0, 5) = 5.339%

Exercise 4.3 Suppose we know from market prices, the following zero-coupon rates with matu-

rities inferior or equal to one year:

Maturity Zero-coupon Rate (%)

1 Day 3.20

1 Month 3.30

2 Months 3.40

3 Months 3.50

6 Months 3.60

9 Months 3.80

1 Year 4.00

Now, we consider the following bonds priced by the market until the 4-year matu-

rity:

Maturity Annual Coupon (%) Gross Price

1 Year and 3 Months 4 102.8

1 Year and 6 Months 4.5 102.5

2 Years 3.5 98.3

3 Years 4 98.7

4 Years 5 101.6

The compounding frequency is assumed to be annual.

1. Using the bootstrapping method, compute the zero-coupon rates for the follow-

ing maturities: 1 year and 3 months, 1 year and 6 months, 2 years, 3 years and

4 years.

2. Draw the zero-coupon yield curve using a linear interpolation.

Solution 4.3 1. We first extract the 1-year-and-3-month maturity zero-coupon rate. In the ab-

sence of arbitrage opportunities, the price of this bond is the sum of its future

discounted cash flows:

102.8 =4

(1 + 3.5%)1/4+

104

(1 + x)1+1/4

where x is the 1-year-and-3-month maturity zero-coupon rate to be determined.

Solving this equation (for example with the Excel solver), we obtain 4.16% for

x. Applying the same procedure with the 1-year and 6-month maturity and the

2-year maturity bonds, we obtain respectively 4.32% and 4.41% for x. Next,


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we have to extract the 3-year maturity zero-coupon rate, solving the following

equation:

98.7 =4

(1 + 4%)+

4

(1 + 4.41%)2+

104

(1 + y%)3

y is equal to 4.48% and finally, we extract the 4-year maturity zero-coupon ratedenoted by z, solving the following equation:

101.6 =5

(1 + 4%)+

5

(1 + 4.41%)2+

5

(1 + 4.48%)3+

105

(1 + z%)4

z is equal to 4.57%.

2. Using the linear graph option in Excel, we draw the zero-coupon yield

curve

3.00

3.20

3.40

3.60

3.80

4.00

4.20

4.40

4.60

4.80

0 1 2 3 4

Maturity

Zero-couponrate(%)

Exercise 4.4 1. The 10-year and 12-year zero-coupon rates are respectively equal to 4% and

4.5%. Compute the 111/4 and 113/4-year zero-coupon rates using linear inter-

polation.

2. Same question when you know the 10-year and 15-year zero-coupon rates that

are respectively equal to 8.6% and 9%.

Solution 4.4 Assume that we know R(0, x) and R(0, z) respectively as the x-year and the

z -year zero-coupon rates. We need to get R(0, y), the y-year zero-coupon rate

with y [x; z]. Using linear interpolation, R(0, y) is given by the following for-

mula:

R(0, y) =(z y)R(0, x) + (y x)R(0, z)

z x

1. The 111/4 and 113/4-year zero-coupon rates are obtained as follows:

R(0, 111/4) =0.75 4% + 1.25 4.5%

2= 4.3125%


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R(0, 113/4) =0.25 4% + 1.75 4.5%

2= 4.4375%

2. The 111/4 and 113/4-year zero-coupon rates are obtained as follows:

R(0, 111/4) = 3.75 8.6% + 1.25 9%

5= 8.70%

R(0, 113/4) =3.25 8.6% + 1.75 9%

5= 8.74%

Exercise 4.7 From the prices of zero-coupon bonds quoted in the market, we obtain the follow-

ing zero-coupon curve:

Maturity (years) Zero-coupon Rate R(0, t ) (%) Discount Factor B(0, t )

1 5.000 0.95238

2 5.500 0.898453 5.900 0.84200

4 6.200 0.78614

5 ? ?

6 6.550 0.68341

7 6.650 0.63720

8 ? ?

9 6.830 0.55177

10 6.900 0.51312

where R(0, t ) is the zero-coupon rate at date 0 for maturity t, and B(0, t ) is the

discount factor at date 0 for maturity t.

We need to know the value for the 5-year and the 8-year zero-coupon rates.

We have to estimate them and test four different methods.

1. We use a linear interpolation with the zero-coupon rates. Find R(0, 5), R(0, 8)

and the corresponding values for B(0, 5) and B(0, 8).

2. We use a linear interpolation with the discount factors. Find B(0, 5), B(0, 8)

and the corresponding values for R(0, 5) and R(0, 8).

3. We postulate the following form for the zero-coupon rate function

R(0, t ):

R(0, t ) = a + bt + ct2 + dt3

Estimate the coefficients a, b, c and d, which best approximate the given zero-coupon rates using the following optimization program:

Mina,b,c,d

i

(B(0, i)

B(0,i))2

where B(0, i) are the zero-coupon rates given by the market.

Find the value for R(0, 5) =

R(0, 5),R(0, 8) =

R(0, 8), and the corre-

sponding values for B(0, 5) and B(0, 8).


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4. We postulate the following form for the discount function

B(0, t ):

B(0, t ) = a + bt + ct2 + dt3

Estimate the coefficients a, b, c and d, which best approximate the given

discount factors using the following optimization program:

Mina,b,c,d

i

(B(0, i)

B(0,i))2

where B(0, i) are the discount factors given by the market.

Obtain the value for B(0, 5) =

B(0, 5),B(0, 8) =

B(0, 8), and the corre-

sponding values for R(0, 5) and R(0, 8).

5. Conclude.

Solution 4.7 1. Consider that we know R(0, x) and R(0, z) respectively as the x-year and the

z-year zero-coupon rates and that we need R(0, y), the y-year zero-coupon rate

with y [x; z]. Using linear interpolation, R(0, y) is given by the followingformula:

R(0, y) =(z y)R(0, x) + (y x)R(0, z)

z x

From this equation, we find the value for R(0, 5) and R(0, 8)

R(0, 5) =(6 5)R(0, 4) + (5 4)R(0, 6)

6 4=

R(0, 4) + R(0, 6)

2= 6.375%

R(0, 8) =(9 8)R(0, 7) + (8 7)R(0, 9)

9 7=

R(0, 7) + R(0, 9)

2= 6.740%

Using the following standard equation in which lies the zero-coupon rate R(0, t )

and the discount factor B(0, t )

B(0, t ) =1

(1 + R(0,t))t

we obtain 0.73418 for B(0, 5) and 0.59345 for B(0, 8).

2. Using the same formula as in question 1 but adapting to discount factors

B(0, y) =(z y)B(0, x) + (y x)B(0, z)

z x

we obtain 0.73478 for B(0, 5) and 0.59449 for B(0, 8).

Using the following standard equation

R(0, t) =

1

B(0, t)

1/t

1

we obtain 6.358% for R(0, 5) and 6.717% for R(0, 8).3. Using the Excel function Linest, we obtain the following values for the param-

eters:


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Parameters Value

a 0.04351367

b 0.00720757

c 0.000776521

d 3.11234E-05

which provide us with the following values for the zero-coupon rates and asso-ciated discount factors:

Maturity R(0, t ) (%)

R(0, t ) (%) B(0, t )

B(0, t)

1 5.000 4.998 0.95238 0.95240

2 5.500 5.507 0.89845 0.89833

3 5.900 5.899 0.84200 0.84203

4 6.200 6.191 0.78614 0.78641

5 ? 6.403 ? 0.73322

6 6.550 6.553 0.68341 0.68330

7 6.650 6.659 0.63720 0.63681

8 ? 6.741 ? 0.59339

9 6.830 6.817 0.55177 0.55237

10 6.900 6.906 0.51312 0.51283

4. We first note that there is a constraint in the minimization because we must

have

B(0, 0) = 1

So, the value for a is necessarily equal to 1.Using the Excel function Linest, we obtain the following values for the

parameters:

Parameters Valuea 1

b 0.04945479

c 0.001445358

d 0.000153698

which provide us with the following values for the discount factors and associ-

ated zero-coupon rates:

Maturity B(0, t )

B(0, t ) R(0, t ) (%)

R(0, t ) (%)

1 0.95238 0.94925 5.000 5.346

2 0.89845 0.89654 5.500 5.613

3 0.84200 0.84278 5.900 5.867

4 0.78614 0.78889 6.200 6.107

5 ? 0.73580 ? 6.328

6 0.68341 0.68444 6.550 6.523

7 0.63720 0.63571 6.650 6.686

8 ? 0.59055 ? 6.805

9 0.55177 0.54988 6.830 6.871

10 0.51312 0.51461 6.900 6.869


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5. The table below summarizes the results obtained using the four different meth-

ods of interpolation and minimization

Rates Interpol. DF Interpol. Rates Min. DF Min.

R(0, 5) 6.375% 6.358% 6.403% 6.328%

R(0, 8) 6.740% 6.717% 6.741% 6.805%B(0, 5) 0.73418 0.73478 0.73322 0.73580

B(0, 8) 0.59345 0.59449 0.59339 0.59055

Rates Interpol. stands for interpolation on rates (question 1). DF Interpol.

stands for interpolation on discount factors (question 2). Rates Min stands for

minimization with rates (question 3). DF Min. stands for minimization with

discount factors (question 4).

The table shows that results are quite similar according to the two methods

based on rates. Differences appear when we compare the four methods. In par-

ticular, we can obtain a spread of 7.5 bps for the estimation of R(0, 5) between

Rates Min. and DF Min., and a spread of 8.8 bps for the estimation of R(0, 8)between the two methods based on discount factors. We conclude that the zero-

coupon rate and discount factor estimations are sensitive to the method that is used:

interpolation or minimization.

Exercise 4.8 From the prices of zero-coupon bonds quoted in the market, we obtain the follow-

ing zero-coupon curve:

Maturity (years) R(0, t ) (%) Maturity (years) R(0, t) (%)

0.5 7.500 5 8.516

1 7.130 6 8.724

1.25 7.200 7 8.8462 7.652 8 8.915

3 8.023 10 8.967

4 8.289

where R(0, t ) is the zero-coupon rate at date 0 with maturity t, and B(0, t ) is the

discount factor at date 0 with maturity t.

We need to know the value for R(0, 0.8), R(0, 1.5), R(0, 3.4), R(0, 5.25),

R(0, 8.3) and R(0, 9), where R(0, i) is the zero-coupon rate at date 0 with maturity

i. We have to estimate them, and test two different methods.

1. We postulate the following form for the zero-coupon rate function

R(0, t ):

R(0, t ) = a + bt + ct2 + dt3

(a) Estimate the coefficients a, b, c and d, which best approximate the given

zero-coupon rates using the following optimization program:

Mina,b,c,d

i

(R(0, i)

R(0,i))2


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where R(0, i) are the zero-coupon rates given by the market. Compare these

rates R(0, i) to the rates

R(0, i) given by the model.

(b) Find the value for the six zero-coupon rates that we are looking for.

(c) Draw the two following curves on the same graph:

The market curve by plotting the market points. The theoretical curve as derived from the prespecified functional

form.

2. Same question as the previous one. But we now postulate the following form

for the discount function

B(0, t ):

B(0, t ) = a + bt + ct2 + dt3

Estimate the coefficients a , b , c and d, which best approximate the given

discount factors using the following optimization program:

Mina,b,c,d

i(B(0, i)

B(0,i))2

where B(0, i) are the discount factors given by the market.

3. Conclude.

Solution 4.8 1. (a) Using the Excel function Linest, we obtain the following values for the

parameters

Parameters Value

a 0.070774834

b 0.00254927

c 0.000175503d 2.44996E-05

which provide us with the theoretical values for the zero-coupon rates

R(0, t )

given by the model and compared with the market values R(0, t )

Maturity (years) R(0, t) (%)

R(0, t ) (%)

0.5 7.500 7.209

1 7.130 7.348

1.25 7.200 7.419

2 7.652 7.6383 8.023 7.934

4 8.289 8.221

5 8.516 8.485

6 8.724 8.710

7 8.846 8.882

8 8.915 8.986

10 8.967 8.932


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(b) We find the value for the six other zero-coupon rates we are looking for in

the following table:

Maturity (years)

R(0, t) (%)

0.8 7.291

1.5 7.4913.4 8.051

5.25 8.545

8.3 9.002

9 9.007

(c) We now draw the graph of the market curve and the theoretical curve. We

see that the three-order polynomial form used to model the zero-coupon rates

is not well adapted to the market configuration, which is an inverted curve

at the short-term segment.

7.00

7.50

8.00

8.50

9.00

0 1 2 3 4 5 6 7 8 9 10

Maturity

Zero-couponrates(%)

2. (a) We first note that there is a constraint in the minimization because we must

have

B(0, 0) = 1

So the value for a is necessarily equal to 1.

Using the Excel function Linest, we obtain the following values

for the parameters:

Parameters Value

a 1

b 0.06865628

c 0.000397845

d 0.000151057

which provide us with the following values for the discount factors andassociated zero-coupon rates:


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Maturity (years) B(0, t)

B(0, t ) R(0, t) (%)

R(0, t) (%)

0.5 0.96449 0.96559 7.500 7.254

1 0.93345 0.93110 7.130 7.400

1.25 0.91676 0.91385 7.200 7.473

2 0.86289 0.86230 7.652 7.689

3 0.79333 0.79453 8.023 7.968

4 0.72721 0.72868 8.289 8.235

5 0.66456 0.66565 8.516 8.480

6 0.60541 0.60637 8.724 8.695

7 0.55248 0.55172 8.846 8.867

8 0.50501 0.50263 8.915 8.979

10 0.42369 0.42471 8.967 8.941

(b) By using the standard relationship between the discount factor and the zero-

coupon rate

R(0, t ) =

1

B(0, t )

1/t 1

we find the value for the six other zero-coupon rates we are looking for inthe following table:

Maturity (years)

B(0, t )

R(0, t) (%)

0.8 0.94490 7.342

1.5 0.89663 7.545

3.4 0.76791 8.077

5.25 0.65045 8.537

8.3 0.48912 8.998

9 0.45999 9.012

(c) We now draw the graph of the market curve and the theoretical curve. We

can see that the three-order polynomial form used to model the discountfunction is not well adapted to the market configuration considered.

7.00

7.50

8.00

8.50

9.00

0 1 2 3 4 5 6 7 8 9 10

Maturity

Zero-co

uponrates(%)


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3. Note first that this is a case of an inverted zero-coupon curve at the short-term

end. We conclude that the two functional forms we have tested are unadapted

to fit with accuracy the observed market zero-coupon rates.

Exercise 4.10 Consider the Nelson and Siegel model

Rc(0, ) = 0 + 1

1 exp

+ 2

1 exp

exp

Our goal is to analyze the impact of the parameter 1/ on the zero-coupon

curve for three different configurations, an increasing curve, a decreasing curve

and an inverted curve at the short-term end.

1. We consider the increasing curve corresponding to the following base-case

parameter values: 0 = 8%, 1 = 3%, 2 = 1% and 1/ = 0.3. We give

successively five different values to the parameter 1/: 1/ = 0.1, 1/ = 0.2,

1/ = 0.3, 1/ = 0.4 and 1/ = 0.5. The other parameters are fixed. Draw

the five different yield curves to estimate the effect of the parameter 1/.

2. We consider the decreasing curve corresponding to the following base-caseparameter values: 0 = 8%, 1 = 3%, 2 = 1% and 1/ = 0.3. We give

successively five different values to the parameter 1/: 1/ = 0.1, 1/ = 0.2,



3. We consider the inverted curve corresponding to the following base-case param-

eter values: 0 = 8%, 1 = 1%, 2 = 2% and 1/ = 0.3. We give suc-

cessively five different values to the parameter 1/: 1/ = 0.1, 1/ = 0.2,



Solution 4.10 1. The following graph shows clearly the effect of the parameter 1 / in the fivedifferent scenarios for an increasing curve. The parameter 1/ affects the slope

of the curve. The higher the 1/, the more rapidly the curve goes to its long-term

level (8% in the exercise).

0.05

0.055

0.06

0.065

0.07

0.075

0.08

0 5 10 15 20 25 30

Maturity

Zero-co

uponr

ate

1/t= 0.1

1/t= 0.2

1/t= 0.3

1/t= 0.4

1/t= 0.5


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2. The following graph shows clearly the effect of the parameter 1/ in the five

different scenarios for a decreasing curve. The parameter 1/ affects the slope of

the curve. The higher 1/, the more rapidly the curve goes to its long-term level

(8% in the exercise). The effect for a decreasing curve is exactly symmetrical

to the effect for an increasing curve.

0.08

0.085

0.09

0.095

0.1

0.105

0.11

0 5 10 15 20 25 30

Maturity

Zero-couponr

ate

1/t= 0.1

1/t= 0.2

1/t= 0.3

1/t= 0.4

1/t= 0.5

3. The following graph shows clearly the effect of the parameter 1/ in the five

different scenarios for an inverted curve. The parameter 1/ affects the slope

of the curve, and the maturity point where the curve becomes increasing. The

higher 1/, the lower the maturity point where the curve becomes increasing.

For example, this maturity point is around 1.5 years for 1/ equal to 0.5, andaround 8 years for 1/ equal to 0.1.

Maturity

Zer

o-couponr

ate

0.068

0.069

0.07

0.071

0.072

0.073

0.074

0.075

0.076

0.077

0.078

0 5 10 15 20 25 30

1/t= 0.1

1/t= 0.2

1/t= 0.3

1/t= 0.4

1/t= 0.5


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Exercise 4.15 Consider the Nelson and Siegel Extended model

Rc(0, ) = 0 + 1

1 exp

1

1

+ 2

1 exp

1

1

exp

1

+ 3

1 exp

2

2

exp

2

with the following base-case parameter values: 0 = 8%, 1 = 3%, 2 = 1%,

3 = 1%, 1/1 = 0.3 and 1/2 = 3.

We give successively five different values to the parameter 3: 3 =

3%, 3 = 2%, 3 = 1%, 3 = 0% and 3 = 1%. The other parameters are

fixed. Draw the five different yield curves to estimate the effect of the curvature

factor 3.

Solution 4.15 The following graph shows clearly the effect of the curvature factor 3 for the fivedifferent scenarios:

0.04

0.045

0.05

0.055

0.06

0.065

0.07

0.075

0.08

0 5 10 15 20 25 30

Maturity

Zero-couponrate

b3= 3%b3= 2%base caseb3= 0%b3= 1%

Exercise 4.16 Deriving the Interbank Zero-coupon Yield Curve

On 03/15/02, we get from the market the following Euribor rates, futures

contract prices and swap rates (see Chapters 10 and 11 for more details about

swaps and futures)


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Maturity Euribor Contract Futures Maturity Swap

Rate Maturity Price Maturity Rate (%)

03/22/02 4.07% 06/15/02 95.2 03/15/04 5.20

03/29/02 4.11% 09/15/02 95.13 03/15/05 5.36

04/15/02 4.15% 12/15/02 94.93 03/15/06 5.49

05/15/02 4.26% 03/15/03 94.79 03/15/07 5.61

06/15/02 4.34% 06/15/03 94.69 03/15/08 5.71

07/15/02 4.44% 09/15/03 94.54 03/15/09 5.75

08/15/02 4.53% 03/15/10 5.79

03/15/11 5.82

03/15/12 5.84

Note that the underlying asset of the futures contract is a three-month Euribor

rate. For example, the first contract matures on 06/15/02, and the underlying asset

matures three months later on 09/15/02.

1. Extract the implied zero-coupon rates from market data.

2. Draw the zero-coupon yield curve by building a linear interpolation between

the implied zero-coupon rates.

Solution 4.16 1. We first extract the implied zero-coupon rates from the Euribor rates using the

following formula:

R

0,x

365

=

1 +

x

360.Euriborx

365x

1

where R 0, x365 and Euribor

xare respectively the zero-coupon rate and the

Euribor rate with residual maturity x (as a number of days).

We obtain the following results:

x Euriborx (%) R

0, x365

(%)

7 4.07 4.211

14 4.11 4.252

31 4.15 4.290

61 4.26 4.398

92 4.34 4.473

122 4.44 4.569

153 4.53 4.654

We now extract the implied zero-coupon rates from the futures price using

the following formula:

Bf

0,x

365,

y

365

=

B

0, y365

B

0, x365


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which transforms into1 + (100 Futures Price)%.

y x

360

=

B

0, x365

B

0,y

365

and finally enables to obtain

R

0,

y

365

=

1

B

0, x365

.1 + (100 Futures Price)%.y x360

365y 1

where B(0, t ) is the discount factor with maturity t and Bf(0, t , T ) the forward

discount factor determined at date 0, beginning at date t and finishing at date T.

Using the last equation, we obtain the following results (FP stands for

Futures Price)

x y B

0, x365

F P R

0, y365

(%)

92 184 0.98903 95.2 4.714

184 275 0.97705 95.13 4.819

275 365 0.96516 94.93 4.923

365 457 0.95308 94.79 5.016

457 549 0.94056 94.69 5.096

549 640 0.92797 94.54 5.175

Detailing the calculations for the first line of the previous table, we obtain

x = number of days between the 03/15/02 and the 06/15/02

y = number of days between the 03/15/02 and the 09/15/02

B

0,x

365

=

1

(1 + 4.473%)92/365= 0.98903

F P = 95.20

R

0,y

365

= 4.714%

We now extract the implied zero-coupon rates from the swap rates using

the following formula:

SR(n)

1 + R(0, 1)+

SR(n)

(1 + R(0, 2))2+ +

1 + SR(n)

(1 + R(0, n))n= 1

which enables us to obtain

R(0, n) =

1

1 SR(n)1+R(0,1) SR(n)

(1+R(0,n1))n1

(1 + SR(n))

1n

1

where SR(n) is the swap rate with maturity n.


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We then obtain the following results:

n SR(n) (%) R(0, n) (%)

2 5.20 5.207

3 5.36 5.374

4 5.49 5.5125 5.61 5.642

6 5.71 5.753

7 5.75 5.795

8 5.79 5.839

9 5.82 5.872

10 5.84 5.893

2. We obtain the following interbank zero-coupon yield curve:

4.00

4.20

4.40

4.60

4.80

5.00

5.20

5.40

5.60

5.80

6.00

0 1 2 3 4 5 6 7 8 9 10

Maturity

Zero-couponrate(%)


Exercise 5.1 Calculate the percentage price change for 4 bonds with different annual coupon

rates (5% and 10%) and different maturities (3 years and 10 years), starting with

a common 7.5% YTM (with annual compounding frequency), and assuming suc-

cessively a new yield of 5%, 7%, 7.49%, 7.51%, 8% and 10%.

Solution 5.1 Results are given in the following table:

fixed income solution ch3

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