Debt and Money Market

Finance 473 Debt and Money Market

Problem set 3

Andrei Simonov

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1. Consider a portfolio consisting of 4 different bonds with market values as in the Table below.

Bond A B C D

Mod. Duration (years) 2 7 8 14

Market value($) 13 27 60 40

a. What is the modified duration of the portfolio? Show that the modified duration of a portfolio of bonds is the (portfolio) weighted average of the individual bond durations.

Modified duration of the portfolio：13+27+60+40=140

D=(13/140) x2+(27/140) x7+(60/140) x8+(40/140) x14=8.96 years

b. Suppose interest rates falls by 50 bp for all maturities. Compute an approximate percentage change in the value of the portfolio.(10p)

Dp/p*100=13x1.01+27x1.035+60x1.04+40x1.07=146.28

Percentage change=(146.28/140)-1=4.5%

2. Consider the two semi-annual bonds E and F described in the table below.

Bond E F

Coupons 8% 9%

Yield to maturity 8% 8%

Maturity (years) 2 5

Face value $100 $100

Price $100 $104.055

Now assume interest rates increases by 100 bp.

a. Calculate the price of the bonds.

Price E: P=4*(1-1/(1+0.09/2)^4)/(0.09/2)+100/(1+0.09/2)^4=$98.2062

Piece F: P=4.5*(1-1/(1+0.09/2)^10)/(0.09/2)+100/(1+0.09/2)^10=$100

b. Approximate the price using duration.

E: PV1=4/(1.04)=3.85

PV2=4/(1.04^2)=3.7

PV3=4/(1.04^3)=3.56

PV4=104/(1.04^4)=88.9

D=(1*3.85/100)+(2*3.7/100)+(3*3.56/100)+(4*88.9/100)= 3.7753

D*=3.7753/1.04=3.63

dp=-(D*)*dy*P=-3.63*0.005*100=-1.815

P*=100-1.815=$98.185

F:=4(1/1.04^2+1/1.04^3+1/1.04^4+1/1.04^5+100/1.04^5)*(-0.005)=-1.82P(E)=100-1.82=98.18P(F)=104.055-1.82=102.235

c. Approximate the price using both