Question

Question 5

Portfolio A consists of a one-year zero-coupon bond with a face value of $2,000 and a 10- year zero-coupon bond with a face value of $6,000. Portfolio B consists of a 5.95-year zero-coupon bond with a face value of $5,000. The current yield on all bonds is 10% per annum (continuously compounded).

5.1 Show that both portfolios have the same duration.

5.2 Show that the percentage changes in the values of the two portfolios for a 0.1% per annum increase in yields are the same.

5.3 What are the percentage changes in the values of the two portfolios for a 5% per annum increase in yields?

Answer 1

Percentage change =(old value-new value)*100/old value

5.1 We need to find portfolio A duration

Portfolio A is a combination of one year zero coupon bond and 10 year 0 coupon bond

For one year bond

T1=1 year ,face value FV1=$2000

For 10 year bond

T2=1 year,face value FV2=$6000

RATE=10%=.1

Therefore Duration of portfolio A=

==(1810+22200)/(1810+2220)

=5.95 years

This is same as duration of portfolio B

5.2

Value of portfolio A when yield is 10%

=FV1*(e^(-yield*T1)+FV2*(e^(-yield*T2)

=2000*(e^(0.1))+6000*(e^(-.1*10)

=$4016.95

When yield increases by .1 percent

Yield=.1+.001=0.101

Therefore portfolio A value becomes

=2000*(e^(.101))+6000*(e^(.101*10))

=$3993.18

Therefore percentags decrease=(4016.95-3993.18)*100/4016.95

=0.59%

The value of Portfolio B when yield is 10 percent

=FV of B*e^(-yield*duration)

=5000*e^(-0.1*5.95)

=$2757.81

When yield increases by 10 BP or yield=0.101

Value of portfolio B=5000*e^(-0.101*5.95)

=$2741.45

Percentage decrease in value of portfolio B

=(2757.81-2741.45)*100/2757.81

=0.59%

Therefore percentage change in value of both portfolios is same.

5.3

If yield increases by 5 percent or 0.05 the. New yield=10 percent plus 5 percent =0.1+0.05=0.15

New Value of portfolio A=2000*e^(-0.15)+6000*e^(0.15*10)

=3060.2

New Value of portfolio B=5000*e^(0.15*5.95)

=2048.15

We know when yield is 10 percent

Value of portfolio A=4016.95

Value of portfolio B=2757.81

Therefore percentage reduction in value of A=(4016.95-3060.2)*100/4016.95=23.82%

Percentage reduction in value of B=(2757.81-2048.15)*100/2757.81=25.73 percent