Chapter 11, Problem 2CP

Summary Introduction

(a)

To Discuss:

Altria has issued bonds that pay annually with the following characteristics:

Coupon	Yield to Maturity	Maturity	Macaulay Duration
8%	8%	15 years	10 years

To determine:

Modified duration using the information above

Introduction:

A bond is a security that creates an obligation on the issuer to make specified payments to the holder for a given period of time. The face value of the bond is the amount the holder will receive on maturity along with the coupon rate which is also known as the interest rate of the bond. Yield to maturity is defined as the discount rate that makes the present payments from the bond equal to its price. In simple terms, it is the average rate of return a holder can expect from that bond. Macaulay Duration is a measure of the sensitivity of the price i.e. the value of principal of a bond to a change in interest rates. Modified duration describes a formula representing the measurable change in the value of a security as against the change in the interest rates.

Answer to Problem 2CP

Modified duration of the bond is 9.26 years.

Explanation of Solution

Modified Duration of Bond= $\frac{Duration}{(1+\frac{YTM}{n})}$

Duration of Bond is the Macaulay Duration of Bond.

YTM is the Yield to Maturity.

n is number of coupon periods in a year.

Modified Duration of Bond= $\frac{10}{(1+\frac{.08}{1})}$

=9.26 years

Modified Duration of Bond= $\frac{Duration}{(1+\frac{YTM}{n})}$

Where,

Duration of bond is the Macaulay Duration of Bond.

YTM is the Yield to Maturity.

n is number of coupon periods in a year

Macaulay Duration of Bond= $\frac{(1+YTM)}{YTM}-\frac{(1+YTM)+n(CR-YTM)}{CR[{(1+YTM)}^n-1]+YTM}$

Where,

YTM is the Yield to Maturity.

CR is the Coupon Rate.

n is the number of Maturity Years.

Macaulay Duration of Bond when Coupon rate is 4%:

= $\frac{(1+.08)}{.08}-\frac{(1+.08)+15(.04-.08)}{.04[{(1+.08)}^{15}-1]+.08}$

= $13.5-\frac{(1.08-.6)}{.04[(3.17-1]+.08}$

= $13.5-\frac{.48}{.1668}$

=10.62 years

Modified duration of Bond= $\frac{10.62}{(1+\frac{.08}{1})}$

=9.83 years

Modified duration of bond has increased from 9.26 years to 9.83 years if the coupon rate is 4% instead of 8%.

Therefore, the direction of change in modified duration is upwards.

Macaulay Duration of Bond when Maturity years is 7 years:

= $\frac{(1+.08)}{.08}-\frac{(1+.08)+7(.08-.08)}{.08[{(1+.08)}^7-1]+.08}$

= $13.5-\frac{1.08}{.08[(1.71-1]+.08}$

= $13.5-\frac{1.08}{.1368}$

=5.61 years

Modified Duration of Bond= $\frac{5.61}{(1+\frac{.08}{1})}$

=5.19 years

Modified Duration of bond has decreased from 9.26 years to 5.61 years if maturity years are 7 instead of 15.

Therefore, the direction of change in the modified duration is downwards.

Summary Introduction

(b)

To Discuss:

Altria has issued bonds that pay annually with the following characteristics:

Coupon	Yield to Maturity	Maturity	Macaulay Duration
8%	8%	15 years	10 years

Explanation as to the reason the modified duration is considered to be a better measure than maturity when calculating the bond's sensitivity to changes in interest rates.

Introduction:

A bond is a security that creates an obligation on the issuer to make specified payments to the holder for a given period of time. The face value of the bond is the amount the holder will receive on maturity along with the coupon rate which is also known as the interest rate of the bond. Yield to maturity is defined as the discount rate that makes the present payments from the bond equal to its price. In simple terms, it is the average rate of return a holder can expect from that bond. Macaulay Duration is a measure of the sensitivity of the price i.e. the value of principal of a bond to a change in interest rates. Modified duration describes a formula representing the measurable change in the value of a security as against the change in the interest rates.

Answer to Problem 2CP

The percentage change in price of the bond is just the product of altered duration and the change in the bond's yield to maturity. This is so because the percentage change in the bond price is proportional to the modified duration which is a natural measure of the bond's exposure to the interest rate volatility.

Explanation of Solution

A bond's maturity relates to the total time duration in which the principal amount needs to be paid back. At the completion of the maturity period, the bond's principal is repaid to the bond's owner and the payment of interest gets stopped.

A bond's duration, is more of a theoretical concept and is used for measuring interest-rate sensitivity. Bond investors need to be cautious with regards to the the interest-rate movements. Any increase or decrease in the rates inversely affects the prices of the bonds. This happens because a rise in the interest rates reduces the value of an existing bond. On the other hand, a decrease in rates causes a rise in the bond's value.

Modified duration takes into account the influence of interest-rate movements on the prices of the bond.

Summary Introduction

(c)

To Discuss:

Altria has issued bonds that pay annually with the following characteristics:

Coupon	Yield to Maturity	Maturity	Macaulay Duration
8%	8%	15 years	10 years

Direction of a change in modified duration if:

1. The coupon of the bond was 4%, not 8%.
2. The maturity of the bond is 7 years, not 15 years

Introduction:

A bond is a security that creates an obligation on the issuer to make specified payments to the holder for a given period of time. The face value of the bond is the amount the holder will receive on maturity along with the coupon rate which is also known as the interest rate of the bond. Yield to maturity is defined as the discount rate that makes the present payments from the bond equal to its price. In simple terms, it is the average rate of return a holder can expect from that bond. Macaulay Duration is a measure of the sensitivity of the price i.e. the value of principal of a bond to a change in interest rates. Modified duration describes a formula representing the measurable change in the value of a security as against the change in the interest rates.

Answer to Problem 2CP

The direction of a change in modified duration will be as follows:

1. The direction of a change in the modified duration will be upwards if the coupon of the bond was 4% rather not 8%.
2. The direction of a change in modified duration will be downwards if the maturity of the bond was 7 years rather not 15 years.

Explanation of Solution

Modified Duration of Bond= $\frac{Duration}{(1+\frac{YTM}{n})}$

Where,

Duration of bond is the Macaulay Duration of Bond.

YTM is the Yield to Maturity.

n is number of coupon periods in a year

Macaulay Duration of Bond= $\frac{(1+YTM)}{YTM}-\frac{(1+YTM)+n(CR-YTM)}{CR[{(1+YTM)}^n-1]+YTM}$

Where,

YTM is the Yield to Maturity.

CR is the Coupon Rate.

n is the number of Maturity Years.

1. Macaulay Duration of Bond when Coupon rate is 4%:

= $\frac{(1+.08)}{.08}-\frac{(1+.08)+15(.04-.08)}{.04[{(1+.08)}^{15}-1]+.08}$

= $13.5-\frac{(1.08-.6)}{.04[(3.17-1]+.08}$

= $13.5-\frac{.48}{.1668}$

=10.62 years

Modified duration of Bond= $\frac{10.62}{(1+\frac{.08}{1})}$

=9.83 years

Modified duration of bond has increased from 9.26 years to 9.83 years if the coupon rate is 4% instead of 8%.

Therefore, the direction of change in modified duration is upwards.

2. Macaulay Duration of Bond when Maturity years is 7 years:

= $\frac{(1+.08)}{.08}-\frac{(1+.08)+7(.08-.08)}{.08[{(1+.08)}^7-1]+.08}$

= $13.5-\frac{1.08}{.08[(1.71-1]+.08}$

= $13.5-\frac{1.08}{.1368}$

=5.61 years

Modified Duration of Bond= $\frac{5.61}{(1+\frac{.08}{1})}$

=5.19 years

Modified Duration of bond has decreased from 9.26 years to 5.61 years if maturity years are 7 instead of 15.

Therefore, the direction of change in the modified duration is downwards.