8/22/2019 QF2101 1112S1 Tutorial 4

1/3

AY2011-12 Sem1 QF2101 Tut 4

NG Wee Seng

Email: matnws@nus.edu.sg

Tel: 65164673

1

QF2101 Basic Financial Mathematics

Tutorial 4

Take the face value of bonds to be 100 unless stated otherwise.

Basic Problems

1 A 10-year coupon bond that pays coupons semi-annually at 7% per annum is priced

$103.635 to yield 6.5%. The Macaulays duration is 7.4083. It can be shown that the

convexity at this yield is 65.239. Estimate the bond price when the yield changes to

(i) 6% (ii) 6.7%.

[C = 65.239 ; DM = 7.4083 /1.0325; P(6%) =107.43 approx; P(6.7%)=102.1612

approx]

2 A portfolio containing two bonds, A and B, of market values $600,000 and $400,000

respectively has a duration of 6.7 years. The duration of bond A is known to be 8.5

years. Determine the duration of bond B.

[ 4 years]

3 When continuously compounding is used, the bond price and Macaulays duration of

a n-year bond that pays coupons annually at coupon rate cc are given

respectively by

nn

i

i eceFP

1

and

nn

i

i necieP

FD

1

,

where is the yield to maturity andFis the face value of the bond.

(i) FindP in terms ofcF

, and

.

(ii) Show that DPd

dP

.

[(i)

n

n

ee

ecF

1

)1( ]

8/22/2019 QF2101 1112S1 Tutorial 4

2/3

AY2011-12 Sem1 QF2101 Tut 4

NG Wee Seng

Email: matnws@nus.edu.sg

Tel: 65164673

2

4 A one-year zero-coupon bond is priced at $95. A two-year bond that pays coupons

annually at 8% has face value of $100 and is priced at $104. Compute the one-year

spot interest rate and two-year spot interest rate, and hence find the forward rate 2,1f

[ 8457.5%,2632.5 21 ss %; 6.43% ]

5 Suppose that the prices of four zero-coupon bonds with maturities one, two, three and

four years are $95, $90, $85, $80 respectively. Compute the spot rates js for

4,3,2,1j and forward interest rates 1, jjf forj =1, 2, 3.

[s = (5.263%, 5.410%, 5.567%, 5.737%) , f= (5.556%, 5.882%, 6.249%)]

6 The current one-year spot rate is 3% and the forward rate from time t= 1 to time t= 2

and from time t=2 to time t= 3 are respectively 4% andx%. Findx if a three-year

bond that pays annual coupons at 4.5% and has face value $1000 is priced at

$1016.56.

[4.8]

7 A one-year bond with annual coupon of 5% and par value of $1000 is trading at

$1018.2. A two-year bond with annual coupon of 6% and par value of $100 is priced

at $104.8.

Compute the one-year spot interest rate and two-year spot interest rate.

[ s2 = 0.034845]

8 It is known that the force of interest,100

)(t

t for 50 t . Calculate the time

value of the cash flow stream C = (0, 0, 2, 2, 2, 2) at time t= 3.

[7.828]

8/22/2019 QF2101 1112S1 Tutorial 4

3/3

AY2011-12 Sem1 QF2101 Tut 4

NG Wee Seng

Email: matnws@nus.edu.sg

Tel: 65164673

3

Discussion Problems

1 (Duration of Perpetual Annuity)

The present value,Pof a perpetual annuity that pays $A at t = 1, 2, 3, .. is given by

P=r

A.

Derive the Macaulays duration,D of this perpetual annuity in two ways.

(i) Use the equation P

ddP)(

MD to find MD , and hence findD.

(ii) FindD directly from the definition

1

1

)1(

)1(

i

i

i

i

rA

rAi

D .

[r

r

r

1,

1 ]

2 For fixed coupon rate and yield to maturity, sketch the graph of duration against time

to maturity for a

(i) zero-coupon bond,

(ii) par-bond.

3 By differentiating both sides of the formula

yy

i

i

1

1

0

with respect toy, prove that for any y (0, 1),

21 )1( y

yiyi

i

.

Calculate, to 4 significant figures, the present value and duration of a perpetual

annuity that pays $1 at time kkt ,2 Z+ given that the effective annual interest

rate is perpetually 5%.

[P = 9.756, D = 21.51]