Chapter 16, Problem 23PS

Summary Introduction

(a)

To Discuss:

Supposing the yield to maturity on both bonds increases to 9%.:

1. The actual percentage loss on each bond.
2. The percentage loss predicted by the duration-with-convexity rule.

Introduction:

When specified payments are made by the issuer to the holder for a given period of time due to an obligation created by a security, then that security is known as Bond.The amount the holder will receive on maturity along with the coupon rate which is also known as the interest rate of the bond is known as the face value of the bond. The discount rate due to which the present payments from the bond become equal to its price i.e. it is the average rate of return which a holder can expect from a bond, is known as Yield to Maturity.

A bond's annual income when divided by the current price of the security is known as the current yield.

Convexity is a measure of the curve in the relationship between a bond's yield and a bond's price.

Answer to Problem 23PS

In case the yield to maturity increases to 9%:

1. The actual percentage loss on zero coupon bond is 11.09% and on coupon bond is 10.72%.
2. The percentage loss predicted by the duration-with-convexity rule on zero coupon bond is 11.06% and on coupon bond is 10.63%.

Explanation of Solution

The price of the zero-coupon bond ($1,000 face value) selling at a yield to maturity of 8% is $374.84 which is calculated as below:

Price of Bond = $\frac{Interest}{{(1+K_d)}^1}+\frac{Interest}{{(1+K_d)}^2}+\mathrm{........}+\frac{Interest}{{(1+K_d)}^n}+\frac{MV}{{(1+K_d)}^n}$

= $\frac{0}{{(1+.08)}^1}+\frac{0}{{(1+.08)}^2}+\mathrm{....}\frac{0}{{(1+.08)}^{12.75}}+\frac{1000}{{(1+.08)}^{12.75}}$

= 374.84

The price of the coupon bond ($1,000 face value) with 6% coupon rate selling at a yield to maturity of 8% is $774.84 which is calculated as below:

Price of Bond = $\frac{Interest}{{(1+K_d)}^1}+\frac{Interest}{{(1+K_d)}^2}+\mathrm{........}+\frac{Interest}{{(1+K_d)}^n}+\frac{MV}{{(1+K_d)}^n}$

= $\frac{60}{{(1+.08)}^1}+\frac{60}{{(1+.08)}^2}+\mathrm{....}\frac{60}{{(1+.08)}^{30}}+\frac{1000}{{(1+.08)}^{30}}$

= 774.84

If Yield to Maturity increases to 9%:

1. The actual price of the zero-coupon bond is $333.28 and calculated as below:

Price of Bond = $\frac{Interest}{{(1+K_d)}^1}+\frac{Interest}{{(1+K_d)}^2}+\mathrm{........}+\frac{Interest}{{(1+K_d)}^n}+\frac{MV}{{(1+K_d)}^n}$

= $\frac{0}{{(1+.09)}^1}+\frac{0}{{(1+.09)}^2}+\mathrm{....}\frac{0}{{(1+.09)}^{12.75}}+\frac{1000}{{(1+.09)}^{12.75}}$

= 333.28

The actual price of the coupon bond is $691.79 and calculated as below:

Price of Bond = $\frac{Interest}{{(1+K_d)}^1}+\frac{Interest}{{(1+K_d)}^2}+\mathrm{........}+\frac{Interest}{{(1+K_d)}^n}+\frac{MV}{{(1+K_d)}^n}$

= $\frac{60}{{(1+.09)}^1}+\frac{60}{{(1+.09)}^2}+\mathrm{....}\frac{60}{{(1+.09)}^{30}}+\frac{1000}{{(1+.09)}^{30}}$

= 691.79

Zero coupon bond:

Actual % loss= $\frac{333.28-374.84}{374.84}$

= -11.09

=11.09% loss

Coupon bond:

Actual % loss= $\frac{691.79-774.84}{774.84}$

=-10.72

=10.72% loss

1. The percentage loss predicted by the duration-with-convexity rule of zero-coupon bond is:

Predicted % loss = $[(-Duration)\times \Delta y]+[0.5\times Convexity\times {(\Delta y)}^2]$

= $[(-11.81)\times 0.01]+[0.5\times 150.3\times {0.01}^2]$

= -0.1106

= 11.06% loss

The percentage loss predicted by the duration-with-convexity rule of coupon bond is:

Predicted % loss = $[(-Duration)\times \Delta y]+[0.5\times Convexity\times {(\Delta y)}^2]$

= $[(-11.79)\times 0.01]+[0.5\times 231.2\times {(0.01)}^2]$

= -0.1063

= 10.63%loss

Summary Introduction

(b)

To Discuss:

To repeat part (a), assuming the yield to maturity decreases to 7%

Introduction:

When specified payments are made by the issuer to the holder for a given period of time due to an obligation created by a security, then that security is known as Bond. The amount the holder will receive on maturity along with the coupon rate which is also known as the interest rate of the bond is known as the face value of the bond. The discount rate due to which the present payments from the bond become equal to its price i.e. it is the average rate of return which a holder can expect from a bond, is known as Yield to Maturity.

A bond's annual income when divided by the current price of the security is known as the current yield.

Convexity is a measure of the curve in the relationship between a bond's yield and a bond's price.

Answer to Problem 23PS

In case the yield to maturity decreases to 7%:

1. The actual percentage gain on zero coupon bond is 12.59% and on coupon bond is 13.04%.
2. The percentage gain predicted by the duration-with-convexity rule on zero coupon bond is 12.56% and on coupon bond is 12.95%.

Explanation of Solution

The price of the zero-coupon bond ($1,000 face value) selling at a yield to maturity of 8% is $374.84 which is calculated as below:

Price of Bond = $\frac{Interest}{{(1+K_d)}^1}+\frac{Interest}{{(1+K_d)}^2}+\mathrm{........}+\frac{Interest}{{(1+K_d)}^n}+\frac{MV}{{(1+K_d)}^n}$

= $\frac{0}{{(1+.08)}^1}+\frac{0}{{(1+.08)}^2}+\mathrm{....}\frac{0}{{(1+.08)}^{12.75}}+\frac{1000}{{(1+.08)}^{12.75}}$

= 374.84

The price of the coupon bond ($1,000 face value) with 6% coupon rate selling at a yield to maturity of 8% is $774.84 which is calculated as below:

Price of Bond = $\frac{Interest}{{(1+K_d)}^1}+\frac{Interest}{{(1+K_d)}^2}+\mathrm{........}+\frac{Interest}{{(1+K_d)}^n}+\frac{MV}{{(1+K_d)}^n}$

= $\frac{60}{{(1+.08)}^1}+\frac{60}{{(1+.08)}^2}+\mathrm{....}\frac{60}{{(1+.08)}^{30}}+\frac{1000}{{(1+.08)}^{30}}$

= 774.84

If Yield to Maturity falls to 7%:

1. The price of the zero increases to $422.04 which is calculated as below:

Price of Bond = $\frac{Interest}{{(1+K_d)}^1}+\frac{Interest}{{(1+K_d)}^2}+\mathrm{........}+\frac{Interest}{{(1+K_d)}^n}+\frac{MV}{{(1+K_d)}^n}$

= $\frac{0}{{(1+.07)}^1}+\frac{0}{{(1+.07)}^2}+\mathrm{....}\frac{0}{{(1+.07)}^{12.75}}+\frac{1000}{{(1+.07)}^{12.75}}$

= 422.04

The price of the coupon bond increases to $875.91 which is calculated as below:

Price of Bond = $\frac{Interest}{{(1+K_d)}^1}+\frac{Interest}{{(1+K_d)}^2}+\mathrm{........}+\frac{Interest}{{(1+K_d)}^n}+\frac{MV}{{(1+K_d)}^n}$

= $\frac{60}{{(1+.07)}^1}+\frac{60}{{(1+.07)}^2}+\mathrm{....}\frac{60}{{(1+.07)}^{30}}+\frac{1000}{{(1+.07)}^{30}}$

= 875.91

Zero coupon bond:

Actual % gain= $\frac{422.04-374.84}{374.84}$

= 0.1259

= 12.59% Gain

Coupon bond

Actual % gain= $\frac{875.91-774.84}{774.84}$

= 0.1304

= 13.04% Gain

The percentage gain predicted by the duration-with-convexity rule of zero-coupon bond is:

Predicted % gain= $\text{[(-Duration)}\times \text{(-}\Delta \text{y)]+[0}\text{.5}\times \text{Convexity}\times {\text{(}\Delta \text{y)}}^2]$

= $\text{[(-11}\text{.81)}\times \text{(-0}\text{.01)]+[0}\text{.5}\times 150.3\times {\text{(0}\text{.01)}}^2]$

= 0.1256

= 12.56%gain

The percentage gain predicted by the duration-with-convexity rule of coupon bond is:

Predicted % gain = $\text{[(-Duration)}\times \text{(-}\Delta \text{y)]+[0}\text{.5}\times \text{Convexity}\times {\text{(}\Delta \text{y)}}^2]$

= $\text{[(-11}\text{.79)}\times \text{(-0}\text{.01)]+[0}\text{.5}\times 231.2\times {\text{(0}\text{.01)}}^2]$

=0.1295

=12.95%gain

Summary Introduction

(c)

To Discuss:

Compare the performance of two bonds in the two scenarios, one involving an increase in rates, the other a decrease. Based on their comparative investment performance, explain the attraction of convexity.

Introduction:

When specified payments are made by the issuer to the holder for a given period of time due to an obligation created by a security, then that security is known as Bond. The amount the holder will receive on maturity along with the coupon rate which is also known as the interest rate of the bond is known as the face value of the bond. The discount rate due to which the present payments from the bond become equal to its price i.e. it is the average rate of return which a holder can expect from a bond, is known as Yield to Maturity.

A bond's annual income when divided by the current price of the security is known as the current yield.

Convexity is a measure of the curve in the relationship between a bond's yield and a bond's price.

Answer to Problem 23PS

The 6% coupon bond, which has higher convexity, outperforms the zero regardless of whether rates rise or fall. The convexity effect, which is always positive, always favours the higher convexity bond.

Explanation of Solution

The 6% coupon bond has a higher convexity and it outperforms the zero regardless of whether fall or rise in rates. Using the duration-with-convexity formula this can be said to be a general property: the effects of duration on the two bonds due to any rates change are equal but the positive convexity effect, which is always as it is, is always seen to favour the higher convexity bond. Thus, if there are equal amounts of change in the yields on the bonds; the lower convexity bond is outperformed by the higher convexity bond, with the same initial yield to maturity and duration.

Summary Introduction

(d)

To Discuss:

In view of your answer to (c), determine whether it is possible for two bonds with equal duration, but different convexity, to be priced initially at the same yield to maturity if the yields on both bonds always increased or decreased by equal amounts, as in this example.

Introduction:

When specified payments are made by the issuer to the holder for a given period of time due to an obligation created by a security, then that security is known as Bond. The amount the holder will receive on maturity along with the coupon rate which is also known as the interest rate of the bond is known as the face value of the bond. The discount rate due to which the present payments from the bond become equal to its price i.e. it is the average rate of return which a holder can expect from a bond, is known as Yield to Maturity.

A bond's annual income when divided by the current price of the security is known as the current yield.

Convexity is a measure of the curve in the relationship between a bond's yield and a bond's price.

Answer to Problem 23PS

In view of the answer to (c), it is not possible for two bonds with equal duration, but different convexity, to be priced initially at the same yield to maturity if the yields on both bonds always increased or decreased by equal amounts, as in this example because no one would be willing to buy the lower convexity bond if it always underperforms the other bond.

Explanation of Solution

This condition would not continue for long. If the lower convexity bond results in the under performance of the other bonds, it would not be preferred by the investors. Hence, this will cause a reduction in the prices of the lower convexity bond and it will lead to an increase in its yield to maturity.

Therefore, the initial yield to maturity of the lower convexity bond will be high. The lower convexity will be balanced by the high yield. If the rates register a slight change, the higher yield- lower convexity bond will perform we will display better performance ll. However, if the rates register a substantial change, the lower yield-higher convexity bond will display better performance.