Problem Bonds

(a) Suppose a U.S. treasury bond with a face value of $100 and two years to maturity pays coupons of 2% p.a., payable semi-annually.  Assuming the yield curve in the US is as follows (these yields are nominal annualized yields), calculate the price of the bond.

Problem 1

Fabozzi pointed out that yield to maturity is the interest rate which will make the present value of the cash flows equal to the price (or initial investment). According to this definition, he also noted that the relationship could be described by the formula below,

[pic 1]

where CFt= cash flow in year t, P = price of the investment, N = number of years,  y = yield to maturity . From the formula, we can notice that there are three factors which are Price of the bond, Cash flow in the future and the Number of years of holding the bond, excluding ‘y'. As these three variables are not correlative, when keeping other two variables unchanged, every one of them can lead to a change in ‘y’. Therefore, the changes of these three factors may lead to the changes in the yield to maturity of the bond.

The change of the price of a bond is one of the reasons. The price may be changed by several reasons, such as the credit quality of the issuer, a premium or a discount rate in selling a bond or a change in the yield on comparable bonds. These changes could affect the price of bonds. As a result, the yield of the bond will change. Another reason is the cash flow in the future. We can’t actually know what interest rate is in the future and the reinvestment risk may also affect the cash flow. Because of the unstable cash flow in the future, the yield to maturity cannot be confirmed. Moreover, the number of years of holding the bond can also affect the YTM. Investors do not need to hold a bond to maturity. They may sell them before the maturity of the bond. Therefore, as the number of years of holding the bond can be changed, the yield of a bond will change all the time.

Problem 2

Let us assume that there are two bonds: one is a premium bond with $100 face value, 20%p.a. coupon rate annually and 5-year to maturity, traded at $116.76; another one is a discount bond with $100 face value, 10%p.a. coupon rate annually and 5-year to maturity, traded at $83.24. We also assume that the prevailing interest rate is 15%p.a.

1. Assuming that the prevailing interest rate does not change, at the maturity data, both the premium bond and the discount bond get the same yield to maturity.
2. According to answer A, we know that the premium bond and the discount bond get the same yield, if the interest rate does not change. However, if the price of the discount bond is lower than $83.24 such as $70 and remain premium bond unchanged, we can conclude that the yield of the discount bond is higher than the yield of the premium bond, since prices and yields move in opposite directions.
3. According to answer A, we know that the premium bond and the discount bond get the same yield, if the interest rate does not change. However, if the price of the discount bond is greater than $83.24 such as $100 and remain premium bond unchanged, we can conclude that the yield of the discount bond is lower than the yield of the premium bond, since prices and yields move in opposite directions.

Problem 3

1. Assume that at t=0 the yield curve is instantaneously flat at 6% pa and the investor do not redeem this bond until maturity.

Solution:

The present value of the bond is

[pic 2]

As the 3-year non-callable bond has the same price as the callable bond. Therefore,

[pic 3]

Solving the function, Coupon rate on a 3-year non-callable bond is equal to 8%.

1. Assume that at t=0 the yield curve is instantaneously flat at 6% pa and the investor redeem this bond in year 1 with the price of 101.80.

Solution:

The present value of the bond is

[pic 4]

As the 3-year non-callable bond has the same price as the callable bond. Therefore,

[pic 5]

Solving the function, Coupon rate on a 3-year non-callable bond is equal to 7.34%.

1. Assume that at t=0 the yield curve is instantaneously flat at 7% pa and the investor do not redeem this bond until maturity.

Solution:

The present value of the bond is

[pic 6]

As the 3-year non-callable bond has the same price as the callable bond. Therefore,

[pic 7]

Solving the function, Coupon rate on a 3-year non-callable bond is equal to 8%.

1. Assume that at t=0 the yield curve is instantaneously flat at 6% pa and the investor do not redeem this bond until maturity.

Solution:

The present value of the bond is

[pic 8]

As the 3-year non-callable bond has the same price as the callable bond. Therefore,

[pic 9]

Solving the function, Coupon rate on a 3-year non-callable bond is equal to 8.03%.

Problem 4

1. Assuming the face value of the 20-year bond is $100

 The present value of this bond is

[pic 10]

[pic 11]

As the first bond is a zero-coupon bond the future value in year 3 is equal to Price3. Therefore

[pic 12]

 the annualized holding period yield for the first bond is 1.93%.

1. Assuming the face value of the 7-year bond is $100

The present value of this bond is

[pic 13]

[pic 14]

The future value in year 3 is

[pic 15]

Therefore,

[pic 16]

1. Assuming the face value of the 20-year bond is $100

 The present value of this bond is

[pic 17]

[pic 18]

As the first bond is a zero-coupon bond the future value in year 3 is equal to Price3. Therefore